Technical Memorandum No. 107 004860-2- M AN ORDERED TABLE OF PRIMITIVE POLYNOMIALS OVER GF(2) OF DEGREES 2 THROUGH 19 FOR USE WITH LINEAR MAXIMAL SEQUENCE GENERATORS By K. Metzger, Jr. R. J. Bouwens Approved by _ _do__ Theodore G. Birdsall COOLEY ELECTRONICS LABORATORY Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan for Contract No. N00014-67-A-0181-0035 Office of Naval Research Department of the Navy Arlington, Virginia 22217 Approved for public release; distribution unlimited. July 1972

ABSTRACT This memorandum contains a brief discussion of the practical aspects of generating binary linear maximal sequences. Included is a table of all primitive polynomials over GF(2) of degrees 2 through 19, not including reverses. These polynomials are ordered by the number of nonzero coefficients as well as by degree. This table complements other available tables.

PREFACE The Acoustic Signal Processing Group at Cooley Electronics Laboratory (CEL) makes extensive use of binary linear maximal sequences in its research program. As a result of this, over the past several years, project personnel have, from time to time, been asked to explain how such sequences are generated. The purpose of this memorandum is to supply some information on the practical aspects of sequence generation and to provide the user with a broad selection of potential sequences. However, no attempt is made to describe the theoretical aspects of these sequences. Such material can be found in the references. Appendix B contains an extensive table for use in implementing binary linear maximal sequence generators. A discussion of its use is also included, making this appendix relatively self-contained. The reader who needs to select a sequence generator quickly should be able to do so by referring only to this appendix. The table of primitive polynomials contained in Appendix B was generated some years ago at CEL for internal use. It differs from other similar tables in two respects, (1) only primitive polynomials (those which yield maximal length binary sequences) are included and (2) these are ordered by the number of nonzero coefficients, iv

as well as by degree. This ordering was used to facilitate the search for polynomials that could be implemented using certain configurations of modulo-2 adders (exclusive-or circuits). The tutorial material presented here is an excerpt from a doctoral dissertation being written by one of the authors (Kurt Metzger).

TABLE OF CONTENTS Page ABSTRACT iii PREFACE iv LIST OF ILLUSTRATIONS vii LIST OF TABLES viii LINEAR MAXIMAL SEQUENCES 1 1. Introduction 1 2. Binary Sequences and Generators 2 3. Maximal Length Sequence Generators 10 3. 1 The SSRG 13 3.2 The MSRG 14 3. 3 Comparison of the SSRG and 15 the MSRG 4. Characteristic Polynomials Yielding 17 Maximal Length Sequences APPENDIX A 20 APPENDIX B 22 REFERENCES 104 DISTRIBUTION LIST 106 vi

LIST OF ILLUSTRATIONS Figure Title Page 1 A simple binary sequence generator 2 2 Generator of Fig. 1 implemented using the 3 7400 family of digital TTL integrated circuits 3 Logic symbol definition 4 4 3-stage linear maximal sequence generator 7 [3, 1, O] MSRG 5 3- stage nonmaximal sequence generator 8 [ 3,2, 1, 0]MSRG 6 (a) Simple shift register generator (SFRG) 11 (b) Modular shift register generator (MSRG) 7 Input/output relations of a delay element 12 using the shift operator vii

LIST OF TABLES Figure Title Page I Maximum sequence period L versus 6 number of stages in generator n II Primitive Polynomials (Mod 2) 21 III Distribution of primitive polynomials 27 IV Ordered primitive polynomials of 28 degrees 2 through 19 viii

LINEAR MAXIMAL SEQUENCES 1. Introduction This memorandum is concerned with the methods of generation of binary or two- state waveforms that are implemented using binary linear maximal sequences to select states. These waveforms are composed of equal duration digits where the amplitude of each digit is assigned one of two possible values depending on whether the corresponding bit in the governing sequence is a "one" or a "zero." When standard statistical tests are applied to these waveforms, it is found that the waveforms possess properties that closely approximate those of a two- state waveform made up of randomly selected digits with each state being equally probable (Bernoulli process). Because of these apparent random properties, waveforms generated from linear maximal sequences are often called pseudorandom or pseudonoise waveforms. Waveforms based on linear maximal sequences are easy to generate and have been found applicable in many areas. They have been used in deep space ranging experiments by NASA (Ref. 1), in random number generation (Ref. 2), in pseudorandom noise generators (Ref. 3) and for generating test signals used in acoustic propagation studies in the ocean (Ref. 4).

2 This memorandum is organized in the following manner. Some of the practical aspects of linear maximal sequences are discussed. Two "canonical" forms of sequence generators are introduced and analyzed. A simple analysis shows how the sequences generated by the two forms are related. Two tables of generator connection patterns, which will permit the generation of linear maximal sequences of various lengths, are provided in the appendixes. The interested reader can find additional material on the structure of linear maximal sequences in Refs. 5 through 9. 2. Binary Sequences and Generators A common method of generating waveforms based on binary sequences involves the use of digital networks made up of delay circuits (flip-flops) and modulo-two adders (exclusive-or circuits). The delays and adders are connected together into feedback arrangements called linear sequence generators. An example of such a generator is shown in Fig. 1. Stage 1 Stage 2 Stage 3 Delay Delay Delay Modulo- 2 Adder Fig. 1. A simple binary sequence generator

The symbols used in Fig. 1 are convenient mathematical abstractions of the hardware actually required to build such a device. Figure 2 illustrates how the generator of Fig. 1 would be implemented using industry standard logic symbols and the 7400 family of TTL digital integrated circuits. Q Q Q Q Q Q 7474 7474 7474 C D C DO C D 7486 Fig. 2. Generator of Fig. 1 implemented using the 7400 family of digital TTL integrated circuits

The properties of the delays (flip-flops) and mod-2 adders (exclusive-or) circuits are given in Fig. 3. Q Q A Flip- Flop Exclusive-OR B Delay Delay Mod- 2 Adder A C Before After Clock Clock Pulse Pulse A B C D Q Q 0 0 0 o 0 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 Flip Flp/Deay 1 1 1 Flip- Flop/Delay Exclusive-OR/Mod- 2 Adder Fig. 3. Logic symbol definition

5 When discussing or working with sequence generators, it is often convenient to write the state of the generator in vector form. For example, if in Fig. 1 stage 1 contains a "one" and stages 2 and 3 contain a "zero, " then the state of the generator can be written in shorthand form as (1,0,0). Such a representation is referred to as the content vector. The value of the content vector changes as the generator is clocked and passes through its collection of permissible states. The content vector can also be written as an octal or decimal number. The bits in the binary representation of these numbers describe the state of the generator. A diagram which shows how the associated sequence generator moves from state to state is referred to as the state diagram for that generator. Figures 4 and 5 contain the content vectors and state diagrams for two different 3-stage sequence generators. The generator in Fig. 4 is termed a linear maximal sequence generator. If this generator is initialized using a nonzero content vector, then as it is clocked it will pass through all possible states (with the exception of the all-zero state) that can be represented using a three-bit binary number. If the all-zero state is the initial state, the generator will remain in this state. (In practice, this state is checked for and if it occurs, the generator is forced into another state. ) The maximum period of a sequence generator using n stages of delay is L = 2n - 1. Table I tabulates L for values of n from

2 through 19. Table I. Maximum sequence period L versus number of stages in generator n nl L | n L 2 3 11 2,047 3 7 12 4,095 4 15 13 8,191 5 31 14 16,383 6 63 15 32,767 7 127 16 65,535 8 255 17 131,071 9 511 18 262,143 10 1,023 19 524,287

U1 e U2 U Generator U1 U2 U3 Value I 0 0 4 0 1 0 2 o o 1 1 1 1 0 6 (. 0 1 1 3 1 1 1 7 1 0 1 5 1 0 0 0 0 0 0 0 0 0 Content Vectors z +z+1 Characteristic Polynomial State Diagram Fig. 4. 3-stage linear maximal sequence generator [ 3, 1,OMsRG

Generator U1 U2 U3 Value 1 0 0 4 0 1 0 2 7 (2 o 0 1 1 1 1 1 7 1 0 0 0 1 1 3 1 1 0 6 0 1 1 1 0 1 5 1 0 1 0 0 0 0 0 0 0 Content Vectors 3 2 State Diagram z z + z + z+ Characteristic Polynomial Fig. 5. 3-stage nonmaximal sequence generator [3, 2, 1,O]MSRG

The generator in Fig. 5 is nonmaximal and can operate in any one of the cycles shown, depending on how it is initialized. This generator is included in order to avoid the mistaken impression that all linear generators produce maximal length cycles. The class of nonmaximal generators is much larger than the class of maximal generators. It should be noted that the ordinary n stage binary counter which cycles through 2n states is a nonlinear sequence generator in the context of the algebraic field described using mod-2 addition [ sometimes referred to as GF(2)]. The characteristic polynomials and MSRG notation found in Figs. 4 and 5 will be explained later. An important difference between maximal and nonmaximal generators can be seen by examining the binary stream of "ones" and "zeros" produced at the output of each delay stage. For the maximal generator, the sequence produced by all stages are identical except for a time shift. However, this is not true for the nonmaximal generator. In this memo only linear maximal sequences are considered.

10 3. Maximal Length Sequence Generators There exist many ways in which a collection of flip-flops and exclusive-or circuits can be connected together ill order to generate digital sequences of maximal length. Out of all possible interconnection patterns attention will be restricted to the two canonical forms shown in Fig. 6. (Other canonical forms are described in Ref. 10. ) The generator in Fig. 6(a) is called the simple shift register generator (S.ERG). This configuration is extremely easy to implement using integrated circuit shift register units. The maximum clock rate is limited by the cumulative delay through the mod-2 adders. The particular sequence generated by the FERG depends upon the number of delay stages present and which switches (si) are closed. The generator of Fig. 6(b) is called the modular shift register generator (MSRG). This generator is an extremely convenient one to program on a digital computer. Also, because this generator never incurs more than one stage of mod-2 adder delay, it can often be operated at clock rates greater than those usable with the SERG. The sequence generated by the MSRG depends on the number of delay stages present and which of the switches (mi) are closed.

S- S s S Q Stage 1 Stage 2 Stage n-I Stage n (a) Simple shift register generator (OSERG) I -1 -1 -i mOZ _ Z n-eo 2 n_ SCtage 1 Stage 2 Stage n-l Stage n (b) Modular shift register generator (MSRG) Fig. 6. Two shift register generators

In Section 2 sequence generators were discussed in physical terms; however, their operation can be easily modeled usi ng finite field mathematics and operator notation. Let the state of stage n in a sequence generator at clock time j be written as u (j). The state of stage n at clock time j-1 is then u (j-1). Define the shift operator z such that u (j) = zun(j-1) Using this notation the input and output of a delay unit can be modeled as shown in Fig. 7. (j) z-1 u(j-1 u(j) = z-u(j-1) Fig. 7. Input/output relations of a delay element using the shift operator Closed switches will be assigned value 1 and open switches assigned value 0.

3. 1 The SSRG. Denote the input to stage 1 of Fig. 6(a) as u (j). Then (mod-2 addition) u(j) - sul(j) + - 1un- l( ) + + S1ul(J) But ui(j) = u0(j-i) = z u0(j) Substituting gives u (j) = (sz+ sz(n-) +s z + +z1 u) For convenience define s = 1 so that s u,(j) = (Z-n izi) Fince so u(j) + so u(j) -0 (mod-2) then IV 1 - zn i s + z s z ) u (o i=0h n- i = Multiplying through by z and combining terms gives

14 ( i s- ni) u(j) = i=O 11 n- l0 The term n? s.z n- I i=O is called the characteristic polynomial of the binary sequence generated by the SERG. The properties of the resulting sequence are governed by the properties of this polynomial. 3. 2 The MSRG. Because mod-2 addition is linear, the principle of superposition can be used to determine the form of the characteristic polynomial for the MSRG. Denote the input of stage one as uo(j). (Note that u (j) and un(j) are identical here. ) If the input to stage one is assume to pass through a fictitious switch m, then the contribution to the output of stage n through switch 0 m.i (0 < i < n-1) can be written as (mod-2 addition) i-n (j) Thus n- 1 uo~j =WA; m. zl") u (j) 0 I 0i=

15 For convenience, define m = 1 and let it be the multiplier of u (j) on the left side of the above equation. 0 Since m u (j) + m u (j) o (mod-2) n o n o Then m + 1 m.zin(j) 0.n 1i=O I Multiplying through by zn and grouping terms gives n mi Z u(j) 0 (LO 1 0 The term n L mM.z i=O is the characteristic polynomial of the binary sequence generated by the MSRG. The properties of the resulting sequence are determined by the properties of this polynomial. 3. 3 Comparison of the SSRG and the MSRG. The sequences generated by the LSSRG and the MSRG are the same if s.= m. O< i<n n-i 1

16 In general, however, the state diagrams for the two generators are not the same. It is easily shown that if s. = m. o< i< n 1 1 then the sequences generated by the SERG and the MSRG are the time reverses of each other. When discussing sequences and sequence generators, it is customary to adopt a shorthand notation. Usually this consists of listing either the nonzero coefficients of the defining characteristic polynomial or the closed switches of the associated generator. Careless use of this shorthand notation can cause difficulties since the representations, although closely related, are different. The following convention will be followed when discussing sequences and generators. Parentheses will be used to describe the characteristic polynomial associated with a particular sequence. For example, the 6 characteristic polynomial z + z + 1 is represented as (6, 1, 0) Square brackets will be used to denote switch closures for the generator named by a subscript. For example, the (6, 1, 0) sequence is generated by the [6, 1, 0]MSRG and the [6, 5, 0]sc:RGG In general when discussing sequences it is best to talk in terms of the defining characteristic polynomial.

17 4. Characteristic Polynomials Yielding Maximal Length Sequences It is beyond the scope of this work to describe how the coefficients of the characteristic polynomials are selected to produce maximal length sequences. The interested reader is referred to Refs. 5 through 9. The article by Elspas (Ref. 6) is particularly lucid. In the above references it is established that if the characteristic polynomial, n n-i z +c zn + c z + + c (mod-2) n n- 1 0 is primitive (and thus also irreducible) on GF(2), then the associated generator will generate a linear maximal sequence. Thus the study of maximal length sequence generators is intimately connected to the study of these polynomials. It can also be established that if the set of n + 1 coefficients {ci} yield a primitive polynomial (and thus a maximal length sequence generator), then the polynomial d z +d z n- +... + dlZ + d n n- 1 0 where d. = cn O i n 1 n-1 - -

18 is also a primitive polynomial. For a given form of sequence generator the use of the above two polynomials will yield sequences which are the time reverses of each other. It is common to refer to the polynomials formed by the c's and the d's (when related as above) as being "the reverses" of each other. For the reader's general information, GF(2) is the algebraic field generated when all operations are performed mod-2. An irreducible polynomial is one which cannot be factored. A primitive polynomial is one which divides z + 1 if k = 2 - 1 but does not divide (without remainder) z + 1 for k < 2 - 1. (Primitive polynomials are irreducible but the converse is not necessarily true.) The switch settings on a generator necessary to generate a linear maximal sequence are determined by the coefficients of the associated polynomial. Each primitive polynomial yields a unique binary maximal length sequence. In the case of the MSRG there is a direct onie-to-one correspondence between coefficients of the primitive polynomial and the switch settings used on the generator, i. e., m. = c. O i< n 1 For the S~RG the switch settings and the polynomial co

19 efficients are related as s. = c 0 < i< n 1 n-i - Two tables of primitive polynomials are provided in the appendixes. The table in Appendix A contains one polynomial for each degree from 2 through 100. Using the techniques described in Ref. 9, these sequences can be used as "seeds" to determine others of the same degree. The notation used in this table is the same as the one used in Section 3. 3. Table IV in Appendix B lists all primitive polynomials of degrees 2 through 19. These are ordered in terms of the number of nonzero coefficients as well as by degree. several examples of the use of Table IV are also included.

APPENDIX A Table II contains one primitive polynomial (mod-2) for degrees 2 through 100. This table is from Ref. 11. By applying the techniques described in Ref. 9, these polynomials can be used to determine others of the same degree. 20

21 Table II. Primitive Polynomials (Mod 2)* ( 1, 0) (51, 6, 3, 1, 0) 2, 1, 0) ( 52, 3, 0) ( 3, 1, 0) ( 53, 6, 2, 1, 0) ( 4, 1, 0) ( 54, 6, 5, 4, 3, 2, 0) ( 5, 2, 0) ( 55, 6, 2, 1, 0) ( 6, 1, 0) ( 56, 7, 4, 2, 0) ( 7, 1, 0) ( 57, 5, 3, 2, 0) ( 8, 4, 3, 2, 0) ( 58, 6, 5, 1, 0) ( 9, 4, 0) ( 59, 6, 5, 4, 3, 1, 0) (10, 3, 0) (60, 1, 0) ( 11, 2, 0) ( 61, 5, 2, 1, 0) ( 12, 6, 4, 1, 0) ( 62, 6, 5, 3, 0) ( 13, 4, 3, 1, 0) ( 63, 1, 0) ( 14, 5, 3, 1, 0) ( 64, 4, 3, 1, 0) ( 1, 1, 0) ( 65, 4, 3, 1, 0) ( 16, 5, 3, 2, 0) ( 66, 8, 6, 5, 3, 2, 0) ( 17, 3, 0) ( 67, 5, 2, 1, 0) ( 18, 5, 2, 1, 0) ( 68, 7, 5, 1, 0) ( 19, 5, 2, 1, 0) ( 69, 6, 5, 2, 0) ( 20, 3, 0) ( 70, 5, 3, 1, 0) ( 21, 2, 0) ( 71, 5, 3, 1, 0) ( 22, 1, 0) ( 72, 6, 4, 3, 2, 1, 0) ( 23, 5, 0) ( 73, 4, 3, 2, 0) ( 24, 4, 3, 1, 0) ( 74, 7, 4, 3, 0) (25, 3, 0) (75, 6, 3, 1, 0) ( 26, 6, 2, 1, ) ( 76, 5, 4, 2, 0) ( 27, 5, 2, 1, 0) ( 77, 6, 5, 2, 0) ( 28, 3, 0) ( 78, 7, 2, 1, 0) ( 29, 2, 0) (79, 4, 3, 2, 0) ( 30, 6, 4, 1, 0) ( 80, 7, 5, 3, 2, 1, 0) ( 31, 3, 0) ( 81, 4, 0) ( 32, 7, 5, 3, 2, 1, 0) ( 82, 8, 7, 6, 4, 1, 0) ( 33, 6, 4, 1, 0) ( 83, 7, 4, 2, 0) ( 34, 7, 6, 5, 2, 1, 0) ( 84, 8, 7, 5, 3, 1, 0) ( 35, 2, 0) ( 85, 8, 2, 1, 0) ( 36, 6, 5, 4, 2, 1, 0) ( 86, 6, 5, 2, 0) ( 37, 5, 4, 3, 2, 1, 0) ( 87, 7, 5, 1, 0) ( 38, 6, 5, 1, 0) ( 88, 8, 5, 4, 3, 1, 0) ( 39, 4, 0) ( 89, 6, 5, 3, 0) ( 40, 5, 4, 3, 0) ( 90, 5, 3, 2, 0) ( 41, 3, 0) ( 91, 7, 6, 5, 3, 2, 0) ( 42, 5, 4, 3, 2, 1, 0) ( 92, 6, 5, 2, 0) ( 43, 6, 4, 3, 0) ( 93, 2, 0) ( 44, 6, 5, 2, 0) ( 94, 6, 5, 1, 0) ( 45, 4, 3. 1,0 ) ( 95, 6, 5, 4, 2, 1, 0) ( 46, 8, 5, 3, 2, 1, 0) ( 96, 7, 6, 4, 3, 2, 0) ( 47, 5, 0) ( 97, 6, 0) ( 48, 7, 5, 4, 2, 1, 0) ( 98, 7, 4, 3, 2, 1, 0) ( 49, 6, 5, 4, 0) ( 99, 7, 5, 4, O) ( 50, 4, 3, 2, O) (100, 8, 7, 2, O) (107, 7,., 3, 2, 1, O) (127, 1, O) *Reprinlted with permission of the publisher The Amierican Mathematical Society, from MATHEMATICS OF COMPUTATION Copyright ( 1962, Volume 16, No. 79, p. 369

APPENDIX B This appendix contains two tables concerning the primitive polynomials of degrees 2 through 19 on GF(2). Table III tabulates the number of primitive polynomials (including reverses) of degrees 2 through 19, segregated according to degree and number of nonzero coefficients. Table IV tabulates the actual polynomials which are also grouped on this basis (reverses are not included). 1. Extracting a Polynomial from Table IV Because the coefficients of the primitive polynomials on GF(2) are ones and zeros, it is convenient to treat them as if they were the coefficients in the binary representation of an integer. Such integers are tabulated in Table IV in octal form. In order to recover a particular polynomial, e. g., that associated with the entry 2358 for n = 7, first write the octal value as a binary number. 2358 = 0100111012 The coefficients of the associated polynomial are obtained by associating the right most binary digit with the zero-th power, the second right most binary digit with the first power, etc. The left most 1 is associated with the n-th power. Any zeros to the left of 22

23 the left most 1 are merely fillers and should be ignored. For the given example 7 4 3 2 2358 yields z + z + z + z + 1 or (7,4,3,2,0) Table IV does not contain entries for the reverse polynomials associated with the given entries. (It should be recalled that for a given generator form, the sequences generated using a given polynomial and its reverse are the time reverses of each other. ) For a given polynomial, the reverse polynomial can be found by reordering the coefficients so the the i-th coefficient in the original polynomial becomes the (n - i)th coefficient in the reverse polynomial For example, the table entry 11318 (n = 9) yields the polynomial 11318 - 0010010110012- z9 + z6 + z + z3 + 1- (9,6,4,3,0) The reverse polynomial of 11318 may be found in many simple ways. For example, reversing the base 2 representation gives 9 6 5 3 1001101001 - z + z + z + z + 1- (9,6,5,3,0) An equivalent procedure involves subtracting each decimal digit in the parenthesized representation from n, e. g., (9,6,4,3,0) becomes (0,3,5,6,9)

24 Another equivalent procedure is to read the powers in the original binary representation from left to right, starting with the left most one as the zero-th power. Fince 1001101001 = 11518 is the reverse of the table entry 11318,it is not contained in Table IV. 2. Converting Table Entries to Sequence Generators Select an entry of the desired degree. For example, for n = 9 and 5 nonzero coefficients 10338 might be selected. 10338 = 001 000 011 0112 Reading from right to left and starting with zero, the positions containing ones are 0, 1, 3, 4, 9 The zeros in positions 10 and 11 are merely fillers and should be ignored. The resulting polynomial is written as (9,4,3,1,0) The corresponding MSRG is [9,4,3,1,0]MSRG _1 +1 2 ++ 67 89

25 The "0" serves as a reminder to feed back the output to the input of the first stage and the 191? is a reminder that this is a 9-stage register. The other numbers "1, " "3" and "4" indicate that there are mod-2 adders after the first, third and fourth stages. The SERG corresponding to this polynomial is [ 9, 8, 6, 5, O]SRG [Recall that the i-th switch position for the SFRG corresponds to the (n - i)th polynomial coefficient. ] 1 2 3 a4 5 6 7 8 9 The "t"Q again serves as a reminder to feed back to the input of the first stage and the "9""6 serves as a reminder that this is a ninestage generator. The "5, " "6, " "8, " and "9" are the stages which are mod-2 summed together to obtain the input to the first stage. The generators associated with the reverse polynomial are easily found by subtracting the positions of the ones in the original polynomial from n and proceeding as above. For the polynomial used above, this results in the polynomial (9, 8, 6, 5, O)

26 The MSRG associated with this polynomial is L 9, 8, 6, 5, 0] MSRG 1 2 3 4 5 6 7 8 + 9 The SSRG associated with this polynomial is [9, 4,3, 1,0] SSRG 1 2 3 4 5 6 7 8 9

Table III. Distribution of primitive polynomials Equation Degree Nonzero Coefficien 3 4 5 6 7 8 910 11 12 13 14 15 16 17 18 19 3 1 2 2 2 22 4 0 22 2 0 0 0 6 0 6 2 0 5 4 4 10 12 16 20 44 18 66 42 82 52 152 72 158 7 4 4 28 28 80 86 236 226 470 368 1050 718 1774 9 2 10 50 36 264 338 720 812 2674 2296 6696 11 4 60 140 450 648 2696 2910 10238 13 4 10 66 156 1006 1470 6766 15 6 12 122 284 1772 17 4 24 190 19 Totals 1 2 2 6 6 18 16 48 60 176 144 630 756 1800 2048 7710 7776 27594

2ND DEGRFE POLYNOMIALS# 3 _NONZERO CO FFFICIENTS 73RD DEGREE POLYNOMIALS, 3 NONZERO COEFFICIENTS H --------- ------------ - 0 0 P OCC CbCD 5_THO DEGRE E _POL YNO M IALCS, _3 NONZE RO COSIE FFI C I EN TS -J C — h~~~~~~~~~~~~~~~ - -- - m -o — -

25TH DEGREE POLYNOMIALS, 5 NONZERO COEFFICIENTS C~7 r67 5TH DEGREE POLYNOMIALS, 3 NONZERO COEFFICIENTS 1(3 6TH DEGREE POLYNOMIALS, 5 NONZERO COEFFICIENTS __ 133 147 7TH DEGREE POLYNOMIALS, 3 NONZERO COEFFICIENTS 203 211

-37TH DEGREE POLYNOMIALS, - 5 NQNZERO COEFFICIENTS - 217 235 247 253 313 7TH DEGREE POLYNOMIALS, 7 NONZERO COEFFICIENTS 277 357 THERE ARENO 8TH DEGREE POLYNOMIALS WITH 3 NONZERO COEFFICIENTS__ 8TH DEGREE POLYNOMI ALS, 5 NONZERO COEFFICIENTS -435 453 455 5 15 543 607 8TH DEGREE POLYNOMIALS, 7 NONZERO COEFFICIENTS 537 717

- 49TH DEGREE POLYNOMIALS, 3 NONZERO COEFFICIENTS 1021......... 9TH DEGREE POLYNOMIALS,. 5 NONZERO COEFFICIENTS -- - - - - - - - - - - - - - - ---- - ---- - - - - - - - - 13'3 1055 1063 1131 - 1207 1225 2'43 1423 - 9TH DEGREE POLYNOMIALS, 7 NONZERO COEFFICIENTS 1137 1157 1167 1.175 1257 1267 1275 1317 13331437 1473 151 7 1533 1617 9TH DEGREE POLYNOMIALS, 9 NONZERO COEFFICIENTS 1577

510TH ODEPEF POLYNOMIALS, 3 NONZERO COEFFICIENTS 2011 ___ __ 10TH DEGREE POLYNOMIALS, 5 NONZERO COEFFICIENTS 2033 2047 2055 2145 2213 2305 2415 2443 2503 3023 ___ 1TH_ DEGREE PrLYNOMIALS_ 7 NONZERO COEFFICIENTS__ _ __ _ _.__ _____-_____215 L ____ 2327.234 23~3___________________2_________________2107__ -2 157- 7 i 3 2 I_ __ _2 3 4 7 323,1 7 3?.7-5.- 2 5_7 _255 3-_ _.2 7?.62.7 - 7 M 3067 3117 3133 3427 _ lOTH DEGREE PCLYNOMIALS, 9 NONZERO COEFFICIENTS Z 7.7.. _.2 Z37 7.2 _ 2773. - 3177... l 7. 33 7

-611TH DEGREE POLYNOMIALSS, 3 NONZERO COEFFICIENTS I11TH DEGREE POLYNOMIAL S, 5 NONZERO COEFFICIENTS 4 02 7 7-40 5 3 405 5 4 107 4143 4145 4 161 421 5 4225 4225 4261 441 5 4423 4445 445 1 4505 4603 5007 5023 5025 11TH DEGREE POLYNOMIALS, 7 NONZERO COEFFICIENTS C~.3 4173 4237 4317 4347 4353 4365 4473 4475 4533 4563 4565 44 17 4653 4655 4671 4707 4745 5155 5235 5247 52 23 5263 5265 5463 5513 5607 5613 5623 __ __ 37 6127 6153 6163 6227 6 2 33 6263 6307 6417 6447 6507 7047 11TH DEGRFF POLYNnmIALS, 9 NONZFRO COEFFICIENTS 4577 4767 5177 _ 5337 5 5373 5477 5537 5557 57_ _ 5 65 7 567 5675 5733 574 7 6277 6367 6557 6637 6673 6727 6747 71 37 7237 731 7

-7T HE RE ARE NO) I11TH DEGREFE POL YNCM IALS WI TH 1 1 NONZERO COEFFI C IEN TS T 12THDEGRN H- DEGREE POLYNOMIALS, A 5, — NONZER3-O_ COEFI- CIQENT F F__ IEN 12TH DEGREE POLYNOMI ALS, 7 NONZERO COEFFICIENTS 1 01.73 13 17 5 1 03 53 10437 10473 105 17 10527 10 55 3 10663103 15053 16027 16047 12TH DEGREE POLYNOMIALSP, 9 NON ZERO COEFF I CIENTS _ 13 73 7 11177 11477 12 727 l?273 5 12 7 53 13 2 75 -13663 __ 1 4 35 7147 -- 14717 14+747 1 543 7 15 52 7 1 564 7 15 2 37 16317 17057

12THDEGREE POLYNOMI ALS, 11 NONZERO COEFFICIENTS 13677 155677 THERE ARE NO 13TH DEGREE POLYNOMIALS WITH 3 NONZERO COEFFIC IENTS - - ~~~~13TH DEGREE POLYNOMIALS, 5 NONZERO COEFFICIENTS 2)033 2n9947 20065 20123 20145 20213 20 215 -~20245 20303201 ___ 29341 ____3__ 29415 _ _ 2 042 5 2 _ 294 51 2 50 3 __ 20611 _ _ __ 2103 1 _ _ __21045 __ _ _21103 _ _ _ _ 21 1 _ _ _ _ _ _ _ 211405 2 2 013 2 20 23 22045 22951 2 2 203 23003 2 3905 24007 203 c 241905 2 5 0 3 30007 _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 13TH DEGREE PLYNOMIALS, 7 NONZERO CO EFF I CIENT S 20157 20237 20257 20273 20275 20363 20457 20473 20547253 2')571 2 9(f-2 7 20635 23707 20715 20743 20 761 21 067 71075 213______ 2135 211l7! 2122'7 2 1?2 7 3 2 12 63' ~ 2 1 3" 1 5' 21447 21453 21507215 _______ ____ 21531 2 1 5-6 1 _ 21607 21615 2_'A. 2625 21643 2-16 51 22037 22075 217______ 2215~5 22 2 33 22 2 35 22 26 5 2 2 397 22 3 13 2 2343 22 42 7 22 43 5225 ___ ____~~~~?2 -7l _ 2? 2 252 5 _ 2254 3 2 2269 7 22 61 3 2 2 625 2263 1 ____22705 205______ 2 3 113 21123 2 3 131 232-07 23 2 23 2333)3 2 342 3 235~03 24037247 ______________?240>7 24 14 7 ____24 165 24253 242 55 24 3 15 243 25 ____24 343. ____24417.243______ 24 453 244 65? 451 3 245 25 2 462 3 24703 2 5 017 2503 5 2506 3256 ____ ~~~25115 __ 25443 ____25503 269 17 26053 26 055?6 1l5_____26?13 26 243 207__ __

- 927023 30n57 30117 30147 30323 30417. 30507 30543 31017 31047 M111 3 31 12 3 31407 32033 32347 32207 34027 34047..______ ~_ __ I.__ A~ 3 —1.-t3_! _Dr EG K REO_ L P L LY.NJ UC A I LS_....- 9 N QN-. EO__CDFF__ C. I ENT__ 20677 20737 20773 _. 21177 __21277 21357..__. 21367 _ ___21373 21557 21575 2166 7 21673 21755 21771 22177 22277 22567 22637 22657 22675 2271 _ 2 7227 2 7 77?5 3_27 _ 323 077 7 ___ 1 723 17.3 2 33 6....23 7 23 2 7 5 2333 ___ 235 ___ 23473 23517 23527 23535 23553 23563 23571 23707 23713 24277 24+ 37 2.._ _ _ 24373 _ __. _. 24477 244567 24575 2463_7 2__4657 _____ 24667 24675 24727 2,. 7, 3 2476 5 2515 7 25173 25175 25 327 25333 25 3 5 3 25363 245 25246 7 25555 5 __ 25627 2563 5 _ ___ 256 5 55 _ 25663 _____ 26077 2657 26275 726327 25 34 7 26353 26457 26467 265473 26 53 3 26547 26617 26647 26653 25 743 2711 7 2__ 271 1 3 ____ 27N3 5. 7153 _ 27 217....72 27._ 27253 _ 27263 763. 27427.____ 27 5 1 3 30177 30277 30 357 30537 33573 30667 30 733 30753 30376 31237 3126 7.___._ 31273 31327 _ 31 33 3347 347 _ 31457 ___ 3162731 63.31633 31653 31 70'7 32137 32167 32173 32333 32347 32437 32467 32517 33037 - C7 ____ _.......3073....313.3... _.__ 1. 3.1. 7 3357 _ 34317__.....34517..341.... 34 5437 __7_ 3/4627__ _ 35057... _ 36 03 7 3611 7 13TH DEGRFE POLYNOMIALS,..11 NONZEPO COEFFICIENTS -.............737.2. 7q 7... 25577. 2. 5 25775.26775.... 27337.3... 27373 27375.....27477...- _ 27537_ __ 27657 227717 2 7747 30 777 31767 31773 32577 32757 33357 33567 33717. 33727 34757 34767..35277 -—. 3533 7._.. 35477..... 35557..._ 355567- 70......7077

-1013TH DEGRFE POLYNOMIALS, 13 NONZERO COEFFICIENTS 27777 35777 THERE ARE NO 14TH DEGREE POLYNOMIALS WITH 3 NONZERO COEFFICIENTS...................................... 14TH DEGPEE POLYNOMIALS, 5 NONZ'ERO COEFFICIENTS -- Z, 9 0 5 3- - - - - - - - - - - ----- - - - - -. - - - - - - - 7 - - - - - I-) 5 3 4r97i 1 40123 40251 40415 49423 40503 41015 41141 41205 41411 42 1 3 42103 44023 4493 1 44103 44203 50007 50045 52003 54303 —, 14TH DEGREE POLYNOMIALS, 7 NONZERO COEFFICIENTS 40137 40173 40257 40273 40275 40317 40353 40363 40473 40633 40635 40 547 4 40655 40665 40725 40731 40761 41127 41235 41313 41315 41343 41351 41607 41625 417 11 42163 4232325 42 34 3 42453 42471 42531 42631 42645 42 705 43017 43035 43065 43 1)7 43131 43143 43161 43245 43395 44073 44133 44147 44343 44435 44455 44471 44551 44607 44651 459065 45145 45213 4 524 3 45605 46035 46107 46113 46125 46215 46245 46305 4t' 50 5 47025 47103 47233 50037 50135 503'25 50455 50523 53645 51 01 7.....51055.....51145 51213 51243 51303.....5201 7 52027 52065 5? 107 5 2 21 5 52 603 53015 54063 54243 551 03 56007 56.013 56293 IC

5?ri3 6'30 57' 601 6147 601 3........025.3 6.0.2.63 61063 61 143........62 02 7 61 6 221 3.62 4 13 6330 7 41!7 7 _ 41 357........414 77..... 41 575.......4_1657 41.75 3 41 755 42 33.7...... 42 37 3......2 42 63 7 4266 7 427 27 42 74 7 42 763 43 17 3 4331 7 43 33 3 43 5 17 4 6 7 z.' 37 45 43 751....41+!7_7 ___4456 7 4 471. 5 5.3 453 57 I 5 6. 453,5. -35 3 4 5 c5 5 4 556 3 45 57 1 45 62 7 45 65 3 4567 1 45 713 45 71 5452 -IN 77- 461 75 4 67. 45 3 27 4354643 7 46 5 1-7 46 5655 5A 6 4 55~63 4706 7 4 71 53 4715 5 47 2 33 4732 3 47345 47 435 4752 5503 5'I25 7 5 5.373 3.... ~7 5 3....5075 5.......5076 5 51347 5.1 533 51 63 3.......5163 5...1 51 7 r., 7 52 13 7 952 173 52 2 37 57273 52335 52457 52,+6 7 52655 523 53057.........5~~~~~~53 23...523, __$32 55__ ______532 63 _ -533)7 ~ -5 3315 -53453.. _53.5_ 535. 5 336 2 3 53 703 54 07 7 54 25 7 54+33 5 54 34 7 543 53 54 517 54617542 54 63 5 54 647 5 503 7 55073 -5 50 75 5532 3 55417 55 427 55 453554 551703 56 13 5 5,6313 5646 3 5664 3 57017. 5 7033 57 107 5747 574 3 C.7 5C'3t.......696377..........665 7-....60673.....6 115 7 -6.1.2.3.7....612 6....12-..613 33- _.16 61 40 7 1 5362 7 63 7 62467 6247 3 62 533 63 973 6 5362 63 24 7 63 2 53...6343 3.....6351 3 64167_. 64237...... 64457.....64 553.....64 61 7 6 4 5 7 057 6513 3 6 514 7 6 5 1 5 33 6 522 7 6542 7 65 507 656 057 66127660 65 42 7...67047... 672 07....70 3 17.......70467.......7103 7..71 117.....71 147......72 117....... 14TH-'[EGR~E-E -POtYN'OM'I'ALS, 1 1'NO'N-ZERO-COEFF ICIENT:S................. 42?77..........43577 " 4'3757......457'3-7 4......t+5-773........467-7'3 -.....473 57 47 537 4+755 — 476 z-7667.....4 7. 7 3 47753 5_'277 7. 51,77........521577... 52767 —......53 357 53477___37_ 53 S5 7 53 675 5 372 7 ~ 51377 545 77 547 37 55 567 55 75 3 562 7756 7 _'A73_ —;571 7...7 17 5 __' -57333-. 7.3.47 5753576527.....17.7 7. 52 57 7 62 75 7 62767 673 337 63 537 63 56 7 64377 65 3-77 65 35 7 6 5 7 f-'5'? 2 7 666 7 3 6 71 73 6 725 7.....675 17 6752 7 67 717......7,07 37 707,57 7 13 7!1 35 7 7 2 337 72 56 7 7?_7 17 7 31 37 732 37 73 257 74 35 7 74-55 7757

-121ZTH DEGREE POLYNOMIALS, 13 N0NZ.=PO COEFFICIENTS_ 57 73 7 677-A7 7 17 77 763 77 771 77.....1-5TH -DEGR EE -POLYN'OMi'A LS, 3 N-ONZ'ER R]-COEFF IC IEN TS 15TH DEGREE POLYNOMIALS, 5 NONZERO COEFFICIENTS. 10')2 7! 00 0S;5 100065 100907 100223 100245 100 303 1 00425 100445 104 ~'- o5~ 1)d 6'41 —' 1 1bZ l l,5 1...01211' 102031 1020 43'... 102061 102203 - i'21' 1 3 0 1 1 1O 3902 1! 104111 1 04 20' 3 104405! 104411 1 100 13 1 10031 1 1004511003 i}']2'i1 1-140 95 — 1i 111003 12-0013 -12010'43 1-2.0045 120 103....120 40 3 122003..100............. -5TrH D..EG-REE- POLEYNOM~I"AL-S, 7 NONZERO COE"FF-ICIENTS 10'. t3 7 l:~ 16s7 1i0 0317......10033'5.......o0'34+7 - 100 36e5 l oo0-2 7 - 6 1005 5 s5 i00! 00871 10 )771 3 1 00C7 15 10O1073 101217 lO 1 23 3 1 01313 10133 1 10 1427I013! O'14~53.........1 -d4 63.....10 150 7.....1 O151 5.....101543 ~ i61-55 1...101 561....01 613 1016':,31 1o'0 101721 1 0-2~5 7 10~2067 102147 1 02 1 55 102171 10 22 27 1 02 265 192435 124 1 C;2,6- 3 1 c,~'207 I1P,'251I3 1025,~ 1 102 615 102643 10 266 1 1 02 705 10271 1 13 I C' 3'?5 3 l O 3127 5 1 C 31 3 1 103 145 103P25 1 1033903 10341.3 10 341 5 1 0342 51041! 9'3461 I19)3 605 19 -361I1 1094067 194135 104171 10421 7 104 265 10430 71031 10('4 4 27 104 44 7 10 4 545 104 63 1 104 72 1 10 50171 1 05 125 105 143 1 0514 510,1 1 0520 7..... 1~2 13...10 526 1...105413.....1 0541 5 1 9 5 503 1 0'6047 1 061 07 106 12 5 1 4 10 51 51 1 9'6 20'7 196f.22?3 I),")2 43 1046251 I106 31 1 1 )9640 7 1,6 41 3 106445 1 60

1-1) 611 1 37 01 3 11)3337 1 1 0 067 _ 1 10 075 110 12 7 _ 110195 11015 1 1023 1 11325 5 1 1'2 7 11304 33 113435 110 51 5 1 106 31 110 643 1 10 651 1 11033 114 1237 1 120 53 1 120C6 9 112 11 3 1 12 12 3 1 12 2 23 11 22 31 1 12?45 1 122 51 120 112 42 5 153 11 3 2 3 I113 025 - 113lr.)3 __140 _ 14 2 25 _ 114 23 1 1 14 30 5 __1413 11 44 5 1 1513 11453 5 1 1 53415 11 60 25 1 16 205 11 64 03 1 17303 1 23305 7103 2"l2 47 12 015 3 - 12315 5 _ 12 02 65 1 20C3 15 _ 120 417 - 1 2 043 9 1 20 44 7 _1?43_ 1 22465 137 125 1 1231712 1 107 1 21 143 1 2 1305r 1 21 407 12?1 4 13111 1214. _ 122 1 23. 1 2 2 2 13.__122 245.2303 -122?4'3~ _1 2260)3-,___ 1231'3 A__ 24 05 3.. 203____ 1 24 11 3 1 2 411 5 1 241 25 1 2 4 223 1242 43 12441 3 1 2442 3 1 253 15 12 5 103 150 1L2620 7 1 2 635 1 2l-001 7 _ 1 3303 3 __13 00 55 __ 1302?23 _ 1 3 03 03 1 3040 7 1 3341 3 111 1 31 3?l3 131 13103 1~2 103 1 3443 3 1 4(3037 1431 153 1 4322 7 1 40 2 47 143313 1 4034 142463 __ _ 143023. 44~~~~~~~~~~~~~~~~~~~~~703 ____..1413?7.. 142063 142137 142113~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ l42~~~~~~~3 _ _*A42413 ____ 144033.. __________~~~~~~~4 -- 421 3142?7 12,1?0 1 53227 1 50 2 13 1 5 1007 16 0 1 7 1601 07 1 5TH DEG REEF POL YNOM IAL S, 9 NONZERO COEFFICIENTS_____ ___ ____J3-3f-.13075.11177 101537 —~ —— 101657. --— 10167 —-10233740 3 71-02-37-5 u257 12?5 73 1302733 1 3,2 75 5 1 02 763 10 3 175 1 03 2 57 1 03 267 103 333 103 553 131 103&53 133655.~~~~~~~ 103663 - 10376> - 4657..il~~~ —--— 0 4717. A104735 1075__105173__153 1 3~529 7 1In5 33 5 10 54 73 1095 595 10 5 563 10 563 5 10l5 71 3 1 05 725 105745 156 1> )2 7 3 106 53 3 16 6 33 10)5 713 _1 01067 15 _ 1067 45. 10 67 51 In713737 I_ n 107 373172 1371 53 1 07 16 5 1 072 17 10 72 27 1 072 33 107323 107345 107417 107447 175 ___..~~~~~~ilQ)7 471 I __1754 5 1 7645 - 107 7 11 11 0573 11 06 7 5 ___ 1107 1 7 ___ 1 1 137 7 ___ 1 1 17 3-__116-_____ 111335 ~~~~1 1 1365 1 113 71 1 11 5?27 11 156 3 1 1163 5 1 1 1665 1 1171 3 1 12?175 127 I 2 7 1 1 2 33' 112 3 47 1 123t65 - 1 12 479 - 112 553 -1 12 6 33 1 12 655 _ 112751 136 I1 13375 11 314 7 I113 171 1 13 22 7 1 1-3,2 55 1 13 3 13 11 34 53 1 13 465 1 13515 1 1354 ____-. 11~~~~~~~~364 3. 11 364 5 -.-1142 57 11 42 73 1137..143. 1445 7 1 1446!.155-__ I1' 6 33 1 14 671 114 7 13 1 147 25 114743 1 1 51 17 1 15 127 1 151 33 1 15 135 155 1 1517 1 1 15 22 7 1 1 52"35 1 1525 3 1 152 71 1 1 5 307. -.. 115 3 43 1 15431.. 1 1 5463 ~152 1 15 61 3 11562 3 1-16()05 7 1 1 6073 1 16 1 35 1 161 47 1 16 165 1 16 1.71 1 163 25 164 116525 11 694 3 1 1661 —3 1 166,4 5 1 17 027 117107. 1171 25 117 14 3 1723143 -- 117 4 23 1 17443- 12-3337 1 20(373? 1 2O~;5 7 1 2 5 73 1 20 63 7 1 20 71I7 12 1 157 112 I121 3 33 I 1235 5. 12 1553. 15~53. 21617____.12 1b27-____ 121 663 -121. 71 5-.___ 1 21I7 2 5- 125 ____ 1 L 2227 5 1 2 2_3 33 1 2 23 55 122-51 7 12 264 7 1 2 27 15 1 230 57 1?23 1~7 1 23 163 133 IL12 235 1 23?'~n7 1 23 4 33 1 23 4 53 1 23 61 5 1 24 175 1 2 4 237 - l247 A 7 1 2 43 17 235 I 2- 55 124 64 7 1 2 46,55 1 2 5097 12 51 17 12 52 35 1 2 52 55 1 25 323 12 5417 153

-141 25465 129 53 7 1-75543 12561 3 126067 126133 126155 126 163 126217 726227 12630t7... 125433 1 2,64 3 - 27-03-5 127055 127143 127215 127243 127423 127503 I 3,0777 1 3016 7 1r 257 1i 3063 1304.75 133527 130535 130617 130627 1 30335 137'97 13'11-7 1...12133 13142,7 131447 131623 132127 132155 132247 132435 132 53 7 137 52' 2 70 7 133.027 133033 13304 7 133063 13311 3 1 33213 133603 1 3' 117 13R4235 1'4323 13'+447 134453 134507 134703 13520'7 135273 13S443 ~5523 1,''hX 13621 - 13630 1 _6423 136603 137043 137233 140373 140477 1 4-3 53 7 4 7!4573 140'793 149747 14152 7 141 533 14156 3 141 633 1413707 14i 743 1?-137 142157 142237! t,4.267 1423535 _ 142457 142533 142633 314 266 i- 3 15 3 14321 7 143227.. 1'43253....143307 143523 144137 144437 144467 144473 144533 ]. 41 7.7 144713 145057 145147 145217 145427 145433 145453 145637 140233 14-247 14 6263 146453 1465607 147047 1'50077 153 167 150317 15032 7I 5,4~. 7 I. 5'~5'A 3! 5: 63 3 1 51 057 15 1 1 2 7 1 51 1 5 3 1 51 2 1 7 15 1427 1 5144 7 1 520 73 I-5?!53..l....52227...152253 1 52+i 7....152147 1525P7..153047 154127 15427 1 54233 1 5424 7 1 5459C7 I_ _ 5 027 155047 157007 _ 163 137 1 60'2 67 160327 160 517 16547 1-111 7 161147 161417 16203 7 16_227 162427 164057 164417 170037 17005 7 15TH DEGREF POLYNOMIALS, 11 NONZFRO COEFFICIENTS 10" 3 37 7 1 3 6 77 103737 104777 106577 106677 10 67 57 107277 107373 107567 107575 111577 111737 11175'7 11 i 773 1-i2 37 7 1127'37 1 13-337 1i3373 3637 ll 3 73 3 1! 735 ] 13763 114577 114677 115177 1 115373 1 15 537 115557 71155b7 115S6'7 115675 115753 116277 116367 1'6637 h116675 117237 117355 117363 11745 7 117517 117547 117633 11 7655 117 63 117665 117715 1 23375 12353 7 23673 67675!237 17 1 23 735 123747 124757 124 767 2.25673 125727 125733 125 73 12 533 7 1 263 75 1 26557 126657 12S735 126747 1 26 753 1 27077 12713 7 1 27167..127?73 127327 127335 - 12734-7 1273'65 - 127437 127457 127473 l27S47 127617 127653 127723 127743 139737 1311.77 131337 131 357 131367 131573 131 667 13!'727 131733 1 31 735 13? 177 132 2'77 132357 132 367 132477 132567 1 32573 132673 lA 27 27 132 747 132 753 133 173 1 33257 133347 1 33,53 133 437 133 52 7 1 3 3 5' A 13 36 35. 1 3 6 3 1'31427i7 - 13"4'37 3 i345h67 1341657 134673 134763 175267 13532 7 135475 135533 135547 135713 1.6 173 136237 136457 136547 16723 1.370. 1377 137127..137133 137153 137253-3 137447 137463 13713 6 1 37643 1 37703 141757 142377 142677 142737 142757 143557 143637 143727 143733 1 44377 ] 45277 14$535 7 145477 14557 3 145747 146177 146 367 1 4h 373 146637 14671 7 1467 7 146753 147237 147773 147327 1 47363 147437 147457 147617!4/707 150677 150757 1515377 -151567 151573 151747 152277 152667 715274r 153 26 7 15332 7 154537 154637 154 72 7 1 55 07 7 1554 37 I 55473 1.5551 7 155707 155 157 15~237 156257. 561 27 1565277 15654+7 157057 157117 157427 160577

-151A)757 161277 161567 161637..161657 161667 _ 163077 163327 163617 164277 I W53 7 164567 164717 164727 166267 166317 167037 167417 170277 170337 1 71157 171237 171457 173037_ 174077 15TH DEGREE POLYNOMIALS, 13 NONZERO COEFFICIENTS _____________________________________________________ 113777 115777 127775 133775 135677 136577 1371i77 137667 146777 147377 ___ 147767 1 5 3 5 7 153677.1 53767 1 5 3 773 _5 5 7 37 15575 7 15677 67_ 156773 _.1.57557 __._7_ 1 1 777 1 63767 166377 166737 166757 167367 167637 171377 171737 172677 173 57! 73477 1 75337 __ ___...........15TH DEGREE POLYNOMIALS, 15 NONZERO COEFFICIENTS 137777 173777 177377 THERE ARE NO 16TH DEGREE POLYNOMIALS WITH 3 NONZERO COEFFICIENTS 16TH DEGREE POLYNOMIA LS, 5 NON7ERO COEFFICIENTS 2700055 2?r'.71 200123 200641 201025 231331 201045 201205 _ 29122?1 - 201241 292A51 2022'03 202211 23? 22 1 202301 2q401 5 204141 210013 21 3133 210233 21J? 5 212? 1 220405 241033 250003 30002 3

16TH DEGREE POLYNOMIALS, 7 NONZFRO COEFFICIENTS.............,,,.........,,,.... 200077 200275 200327 200457 209475 20051 7 200535 200627 200655 200707 201 357 20! 13 ~ 2~'! 155 201345 201435 201513 201 551 201 561 201607 201 61 5 2C:1543 202!27 202147 202453 202507 202551 202607 202 703 203113 203123 2~3 213 203303 2 n3603 20432 5 204 343 204351 2,94471 204613 204645 20470 3 29'- 705 205 37 5 2C5423 205425 20 55~ 1 20601 7 206035 206 151 206161 206215................................. 2',;5225 226 243 206503 206603 207 105 207205 207405 207 41! 21 O1 1 7 210135 2197~7 2194~,5 2 ~0 711 211 123 211143 21121 3 211305 212033 212115 212207.......................... 21272 3 212 3!! 212321 212407 21 3103 214215 214223 214 251 21444 3 214445 215057 215211 215411 216051 216205 220075 220217 220233.. 220271.........220307'. _. 2 2'" 6C, 7 929625 2?065 I. 22! 047 22] 065 221123 2212 13 221223 221 451 221505 222 O17 2? 27 5 5 2 22 071 2 2211 3 222231 222 42 5 223007 2230 25 223 943 2 23203.. 2;~t*-33 5' 224251 224305 224413............224431...... 22450:~...... 224505 224603 225'345 225205 23302 7 2 39C4 7 2?03,9 3 239407 230415 231405 232025 232 045 2 3240 5 2 34015 24) 165' - 24,9235 240315 24n513.......... 240523............243623..........241113 241413......... 241503 241603"242'017 247!~7 242225 242243 2Z, 7305 2424!3' 2440 35 244065 244303 244413 2-i~~ —25~-'245223.......24502~....259017~ 25n40'7 251013 252203 254007 ~'260143....260413 __~ ~'3qO 3 26! 02 3 262007 2 6~_ 103 264~ 13 266003 3004 27 30.0 453 3 O1 033 301213....... 30122 3.......... 3 0149 7 3C'2033 302053.....30 2423 ~30301 3....304063 304 143 31001 7......31 0033...... 3!044 7 31S 107 314.91 3 320407 hm, ~,0 16TH OEGR-EF —POLYNOMIALS, 9 NONZERO COEF-F-[-C-IENTS............ --—,...........,.. --.......,. —................. ---- —, q.. -.-....... 20-T9-~:?-7 -- 201637 201735 201771 202277 202735 20-2:?-53 — 2-0'3'55 3 2'0'35-5'5 — 203571 ~.r-~ 761 2 042 77 20 43 37 2043 73 2 045 75 204657 204 747 20476~ 2 0,5! 75 20 52 73 - 205535.........205'~i 5.........205'723...............2'05745 2061-73 2-0-6-25:7 296 363 20661 7.........20672'3 206745 207235 207247 207433 20755 1 207607 207625 207661 210177 210557 210733 213753 211!67 21'i257 —21-i3'33....21'13-55 —— ~ 2-11'~363 211473 2'il 475 211 565'~ 211571 211635 21225'! 212355 212371 212617 21~65~ 212725 213037 213067 213075 "213233.........21-3'253 -213331 — 2i3507 21352'3 — 21363-1 — 213'643......213551 214157 214273 2'437I 214553 214565 214571 2146~3 214745 215127 215153 215343 21 5345 -— 215453.......2I 5~.07 21 55 13...........2157,33 216235 2152'55 216313......216315'-~21635 1....2.16361 216427 21 6523 216~31 2t.6545 216607 2-17231 217303 217451 217593 221257 -221527...... 221555 22174'3............22223"7-..........22234:7. 22'25-27 222535..........222563 222571............222 633"222 745 223,475 223471 223531 223543 22360 7 223645 223 705 2 24l 73 224365 22453'5....... 224571 224707......... 224725 225073' 22524'7....225315 225'~31........225447..........225523 225631 22~,03 7 22607 3 22611 7 226135 225153 226163 226171 225 27 I 22634 3 276/-5 5 22 654 5 227017 22710 7 227113 227! 15 227305 227503 2 3015 7 230 257 27)3~5 233'~5~ 2~07.65 2 %n553 230647 2'39653 230715 230 731 231 057 231967 231075 231 L27 231133 231155 2312~5 ~ 231265 231331 231433 2~1471 231705 232 23'5 232435 23251 3 2 32643' 233933 233065 233303 233407 23341 3 2 33503 ~34037 2 3c, c, 5 7 2'%41 35 234155 234307 234313 234515 234.q~_3' 234545 235033 235125 23~ 1 z, 5 235'~,35 234! 07 23611 3 235223 236407 236423 236443 236503

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_ o. 2 7~ 33 3 27-~9..~'~ 274627 2 7503 7 275 247 275253 275543 276 967 27611 7 2 76313................... 27',~_ 3......... 2755')7 276637....277053......... 2771 1'3.......... 277213 277223 277413 301377 301577 ~:'! 73 7? r;:~ 577 3m 277 3 303357 303657 304577 304757 3 01'+ 7?3 305667 30 5733 3"'~'~ 57.......~,05 7'* 7 3C7527 307723 311637....... 3! 173'~ 311753 _~1I 763 313137.... 313157 ll ~.7 313?.3 3 31343 7 3 13553 315157 315327 315467 315533 3 I571 3 3 15.~ 17 3171I 7 2! 7 127 3i 7247 317307 321277 321 71 7 322277 322667 322673 323257 372 t*~ 7 ~P3 547 ~..2?.527 ~243'~ 7 3?4657 324747 325 167 325 237 32543 7 32552 7 3256'+7 3_~- I~7 326=~17 326517 326533 327227 327233 330177 330357.... 330557 ~.'~') 673 3 =. 371,7 33 ] ~:~3 33151 7 331617 332 ~57 332473 332707 3331 17 334347 334 z- 57'~3512 7 335227 33544 7 336147 337027 341337 341 637 34166 7 342567 342.657 ~44 ~ 7 344537 344557 345347 345437 346157 346467 34652 7 347127 353,:,57 35072_7 3c, 2457.....354437...........35~517.........355417.....356057 360177 360337........361137 3~.~3.357 364!57 3~5037 37.3137 16TH DEGREE POLYNOMIALS, 13 NONZERO COEFFICIENTS.......__ 2!6777 217677 227377 234777 23-5""T-57 237337 — 2-'37-3-73 245777 254777 2 5-573 7 25.~577 256775 257675. 263677 263737 263757 265773 266.677 267 177 267375, 7_,_ 27~37'273657 27457:7 274757 275357 2756:73 276'277 27672-7 277257 W:~ 277273 __ 2 771,5 7 277653 277737 277723 307757 314777 315577 315737 315757 r,~ 316 7'~ 7 3! 7 33 7 317573 ~3 i'7667 323677' 32'-6'773 32 733 7 327 657 331 577 332677 333357.... 333557... 333727.......334767 335477 335557 335717 336367 336657 337137 337/,57 3374~? 337647 347277 351-757 352677 352 757 352 767 3~3537 3536673_5(,767 _36~737............ 3~. 3837 363..65.7__~ 365357:~67_63.7 371 1 77 3.71 337 16TH DEGREE POLYNO!~I&LS, 15 NONZERO COEFFICIENTS............................................................................ 26777'7......327777...... 337737..........3-57757............'3677'37'- 3'7-5"5-77..................................

-!917TH DEGREE POLYNOMIALS, NONZ ERO COEFF IC IENT S'OO1! 49 I0~. 1' 4 0CI0 O1 17TH DEGREE POLYNOMi ALS, 5 NONZERO COEFFICIENTS 4-0,55-;5- 7.......~-5 0-.4s.....4n0 063 40,012 5 40oo1 51 40o0215~ 40023'1 400243 400305 ~01 40.)4]5 4, 0 4'. 4 1,0445 400541 40110 3 400701 401015 401043 401 051 415 4 O1 10')5 4120 -A 40'1221 40141]1 40!421 40 200 402 025 40203 1 402103 40 4 4 C' )~2 C, 5... 425 __ 432-05 403 011 404013 40 40!5......404121 4042.21.... 404405 440........... 40)5 - 913 4 1001 5 41 )002 3 41PI025 410051 4100 61 410 105 410111 41r(2034033 4 I-) 4e0 5 4....40, 1! 100 4 14321....420013. - 420r043__ 420061...4203.11.....420403..... 4 29-42 I 4 220C1I1 4 300 03 4 30 0 05 4 40 5 1 44021 I 441003 4 50 003 500023 504 500I0B... 5-0)0 2 05 540003 600013 _.....600043 _ 600103'4'0"... 07 7 4 I,7).-'17-3'....4n-O-25-7...4032 7'3 400365 4{)0475 4 00'5'2-7 — 4C6-547 400555 070 400713...4C)731 _ 400 761 40r1 I0 67........401 1 53 -4"n1 17 1.... 4 01 265........401 30 7........40 13 4 3....41 4 01 42 7 40r',1 4~,~ _, 40!447 4]114 63' 401465 40 1523 401661 ~401 703 401721 423 40,2'~5 7 4 f27"I c- 49~2127..40 2 135. _4021z,7.....402155 40 22 17........402 23 5...40225 3 27 402.-, 3 15 402 ~,433 4 C2435 402)447 4027465 402?543- 402545 402 62 3 402 6454033......._4.095........4'I _ I. 01 13.......4 03 131.......40321 _~___ 4,-403_34R.....403341-........ 403 40 7.... 403451....0511 40 3 521 49~ 3 6 n- 40 C5 05 404 06 7 4'~t*-0 7 5 40412 7 404165 404 27 1 4 0D4 3 2 3 442 4.m3 I -40 3 4r0)443 3 4 04't3 5 40445 5 4r)4 4.67 404525 4 0 5 5 I &O z- 51 A)45I 40)4 604 3 4'+74 I 40~ 5 12'5 40514 5 4q;5 1 5 1 405 23 1 405303 405 - I11 40 5 B 2 1 45 4(7 5 41 5 4C'5 44 5,5 46 1 40 5 505 4 0 61 3...405)4 1 406047 406 107....40561 4 5.....06,'..................40.6 2 15......... 463 41 4"64 37...4.06413 406425 4 065 0 5 4070 ]3 407025 40710340'1 "~~~~~~~~~1 4.06 7 4!.47 41 0]] I? 410]2 4.1 7 4 _ 40 307 41.O~J3..5 410~ 34' 41 03 5]...415 4+23 I)2047 412.0523 412255,3 41225)13 4 1 n623 4124 31 4 1 1'43 4121 01 - -) e 4~~~~~~~~12521 ""f! Z ~ I ],......... 4 ]]. ~ 71.~......~~- 411!105....4111 41 414 7 __.44..... 4141135..... 414....14!...2 3-] 4 1 27 4 14 2 1] 41 474 5 4 147. 1 41 43 03 4 14 40)7 414443 4 14 51 1 41 454 1 4 1 9? 4 1 5." ~.! 41 51 21 4 15203 4 1 6125 41604!. 416061 4 16 11 1 41, 4 1 701 1 4 2 7 4 7 9 ~42,'i!lP_ 7 423) 1 35 4 20)1 4 7 4 20 163 4 23 16 5 4 297 2 1 54225 "- 2)63.......42'?~1 42,.' 9 7 420455 4254.... 42'~~ 5 ",2954 5.... 420703 4237 1 2C7415

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-2117TH OFORFE POLYNO1MIALS9, 9 NONZERO COEFFICIENTS 4:3 7 7 4C 177 4 01 3 3 7 4 1 3 75 4 0156 7 4011~667 40 17 33 401 75 3 402367 427.4'7 7 492557 4 02 6 73 40?795 40f)3 3 27 403333 403 3 71 403457 4035 17 433 4~3355 5 4)327 40 3655 405 42~375 1 40 4 27.7 4043 67 4035404 573 444 415 V7 4 )5 1 73 4n5267 405347 40 53 53 405467 4 05 473 4 05 5'35 40 5 571 454 4'~~~<~ 3 4051) 71;1 60 7 7 4 06'1 67 405~2 6 7 *-147 4____. 045 063 4'- ~4 C/,617 406533 49663 406655 4)6 7?3 406743 4 0674 5 407075 471 4r771'I n7?27 40 7 233 407247 407265 4017 351 4 07 417 4 074 53 476 4'"74 71 4 ) 75 4091 4076()0 7 40 Jol2 3 4n~7651 4977 11 419 17 7 4 1 065 7 406 4 ) 7 33 4 ) 73~5 4130747 41I,)7 65 _41 1177 411 13 7 _411 167 4 11 237 4 112 57 413 411347 4 1137 1 4 114 73 411 54+7 41 15 55 4 1156 3 41 15 65 4 1165 5 4 117 31 425 41I2 23-'7 4 1 231 7 4 12 32 7 *4 1?3 33 412533 __ 425 _412571 412 64 7 4126 55 422 4 1?3I 41?751 41 3 973 41 3 7 5 4I'A1 7 4 13 Vt47 4 13 227 413 2 35 4 13?9 5 4127 41It341 7 4134It6 5 41 34 71 4 13 5 13 413152 5 4 13156 1 4 136 15 413 65 1 414 335 444 41 435 3 I434355 41 t4473 4 149 I;3 4146 35 41464 7 4146 63 414 67 1 414 715 445 it15127 —- 4716 3 ~ 41 5165-. 4152 17 —--— 415 233 —-41P 255 ___415323 ~ ~41533 1 456~147._ _ 4 5 43-13 41 54 9 41547 1 415 515 41 55 43 4 1569 7 41 5625 415 74 1 4160 57 463 4 16 171 41 62 55 - 4 1631 3 _ _416315 __ 41 6345 -- -4 16435 __ —--- 416447 416 453 4 416463 463 4 15 62)3 41 654 3 4 16 711 416741 41 704 7 4 17 6 5 4 17 131 417143 4 172 13 472 4 17 -A5 4 174MA 41 74 61 4 17 59)5 41 7521__4765 _ 4 17611 4077 _ 420367 407 4 20 )557 420 575 4?0657 4 2 05 73 420727 420733 42 1 173 421 23 7 421 273 413 4 2135 3 42 135 5 4 1 3 63 _421 457 4 4215 17 4 215 33 4_ 421 53 5 4 2195,53 _ 4 2171 3 412 4 21 745 42"1 7 91 4 22 237 422273 42 23 17 4223147 422 371 422457 422547 421 422635 422_'?713I 423037 42312 7 4231315+ _ —, 4231i5 __42 31 71 423263 _43 271 431 4 2 33'15 423 325 4 2;3 331 423433 423455 423465 4236 13 423623 423 6531 447 424N1,75441 75 4 42?57 4 4246 7 424457,__ 424473 4__ 43447 5 __ 424547 424S1 7__4273_____ 4;~4 75 1 4 24761 425057 429 1 17 4 251 35 425227 425255 425 34 5 425447 456 425 531 4 2 5 53 425623 425625 _ 425643 _ 42 56 51 426127 _ 4617 42621 "+7 426235 __7 4Th247 425329 426361 ~~~~~~~ ~~~~4 2641 7 4 2643426 45-3 4604255426523 464 42655 1 4?5A3 1 4?o64 3 42 7027 427065 _4 271 133 4 271 23 427225!42 7 24+3 476 42~7 31 1 42 741 5 4274?5 4 27 461 427505 433 1 57 4 3017 3 430273 4 30317 402 420333 4333~~~~~~~~A)1-7 4396A47 4 -AI 1 7 439 7 13 43045430 7 51 431 127 - 431l163 412 431 255 434353 43137 4 31 3 23 4 3 136 1 431 43 5 431447 43 14595 4 345 4 31471 431 5?) 43 19311 431623 4 3 16S45 4 3 216 3 432227 4 32 235 4322'>65 4 32 30 7 421 4 23 3 4312 351 432433 43251 3 4 3 2 515 432545 432607 432 62 5 43 2 631 421 _ 4339 17 4 33 "71 4 331 15 433161 3433 2 13. -il-~33 2 31 4 3 32 51 33 3___442___4353______ 4-33511 43 4 367 4 341 17 4 34 139 4341 55 434255 4 34 265 434345 4 3445 4 3447 43450 7 4 3 4&51 4~4 7 1 4 35Q017 43 5 0-47 435 053 435065 4 -35 12 5 4 3 5 13 14515-',323 43551 4-35443 4 3954 51 4 36033 4303 436 113 4 36 1371 4 362 43460 426'~423 436405 436903 4 3 669 5 4 3 72?)3 4 3721 1 __440357 __440375 44055 7 _ 465 44 )75 3 440763 4 4l1 3 7 441 3 33 441 3 53 441317 1 441 553 441 571 4 416 71 411

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16 6531 7 1 6 6T4.57 1 ~,66711 7 167)277 1670967 16572 517 16742 57 1 675 217 1 703 17 7 1737 1 7 ) 3 537 1 705 177 1 7 C:64 77 17 07 137 17075917 17 1037 7 1 711 2 77 171461,37 1 71 543 7 1767 ________ ~172-147 7 17?1',3 7 1722657 ___17 23 197 1724637 ___1726437 17 30 477 17 31 137 1734437 1753 __ 17 4 51237 1744477 1 7 501 77 1 7601 77 1769277 18TH D)EGREE POLYNO)MIALS, _ 15 NONZERO COEFFICIENTS ___ 1J77775__ 1137 67 7 ____ 137757 1147777 1 157 767 1172777 1173577__ 1 173 757 1174777 __ 157 1175~775 1775#,3 7 I212T67 77 1237767 1263777 1?67767 1 273 7 57 1277337 1277357 1273 12 77 717 177 735 1 315 7 77 131 777 5 1323777 _1326777 1 33 3377 1 333 73 7 1 336 37 7 _1367___ 13 3617 3 7 1 337 27 7 1 3 37 7 7 1347577 1347773 1353757 1355767 1 356 73 7 1 357 17 7 1377 1357763 1 3 5375 7 1 36-5 3 77 1365577 13 66 737 _ 1366773 1367 337 __1367367 ___1367557 1373____ __ 1 3 6771 7 P?7 15 77'A.3 7 1677 13727775 1373637 1373667 1 374 775 1375367 1379537 1357 ____ ~~~~~1S375357 __ l3 7,,6 7 l37~7?7 ___ — 13'-77377 1377527 __1435777 __14 576 77 1457737 1457757 1777_____ I-71- 5S77 ~ 4 77?37 1517 7 37 1535377 1 5317 367 1537537 1 53 7 567 1537573 1537657 1577 1 5 4767,-7 1 547 75 7 1553757 _1556757 1557557 __1557727 1 563 577 __ 1563 773 1567177 _1573____ 15-j 7 S7 3 15576S7 1570777 1573357 15 73 47 7 1573657 1 5737 47 1575477 1575557 1572 _________~~ 575 33 7 1576557 1 57671 7 1576747 __1577257 1577547 ___16 17 737 __1617757 16327771657 1,535757 16347 27 7 163,7337 1637537 1653377 1 6 53 737 165 9677 1657537 1665677 163715' 5577 16b7477'667657 __ 1673277 1674577 __1676637.16771 37 __1677 157 1737 1077____ 1l~71 357 7 1 71 7477 17 17 637 1725577 1727557 173~4737 173 52 77 1736337 17 366371747 ___ ~~~1762677 17731377 18TH DEGREE POLYNOMI ALS9 17 NONZERO COEFFI CI ENTS 1373777 1377773 - 1577677 1637777 1676777 1677577 16 77 737 1737377 1755777 1757 1757677 1766777__ _ _ _ _ ___ __ _ _

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~ "-75'.....:............ i 317~573 3'175~>5 7 ~175753 3175763 3175557 3178573 3176637 3177077 3177Z73 3177333'~! 77473 31 77547 31775 17 3! 776~7 3207777 3~_15777 3217767 321777'3 322.~777 3227577 3727677 3227757......... 32:~3577 5233737........3233773..........3~_35577........3235677............3235677 3235757...........3236737 3237.~57 3237 ~77 32? 7537 3237~.~. 7 3237753 3247677 3251777 3253 ~77 3253 773 3255577 32-~5737 3257 177 32.~' 72 77 3257357 3257373 3257637 3257753 3262777 3263577 32.65737 32 _~5757:~255 7'73 32*-5377 326~,577 3~_65757'~26.~77~ ~287277 3267357 3267567 3267573 32 ~ 772 7 32~ 775 3 327~.7 77 32 71 377 3272577 ~ ~_ 73 ~-37 32 73537 32 73 557 3 ~7~7 17 32 74577 3?75337 327~357 327-~ 37'~ 3275537 32 7 _~.~57 3275733 32767/+7 32'77'377 3277157 3277167'~277!73..........3277~1 7 3277333........327731*7.........3277473 _3277627 ~:~3~7 577 3313757 3314777 3316877 331~,7,57 33 17 37 3 ~31 7657 331773 3 33177/,7 332.~77 332 3577 3325 377 3325577 3326577 3'~27567.......?327573 3327667 332771 7 3327733........3331677 -....3331737....3332737 3332757 3332773 ~333177 3333277 333~567 33 ~47~7 3335373 3335557'3335673 3335 747 333.6 337 3336567:~:~ 35f_,6..7......._~3~7257.............3337 27 3 ~337~1.7....3~Z+7777 ~345 37 7 33/,6677......3347277..........33/+.7537.....3352 767 33 ~.~773'~353 37 3 33'r~ 3~. ~7 33536~7 33~7~_ 7 335~757 3355367 3355477 3 3~5537 3355567 3.~.9~ 57 3 3355637 33c~.~73 3355 373 33~6717 33572:~ 7 3357267 335734 7 33 5745 7 336I 577 33633'~ 7 335 3 373 33~,~57 336~677 3384737 33~4767 33653 37 3365~6 7 33 6571 7 336572? 3366177 33~,~'~'$7 ~367157 3~,67267 3367/.,67 3371477 33 71557 33'71637 3371727 3372537....................................................................... 33 7255 7 3377 567 33727! 7 33 72747 ~373977 3373257.~3 73457 33 7~. 177 3374727 33 75347 3375437.....337~ 7 ~376!~'r.~ ~37~2q7 3376?5.7.........3375:~]7 3376517 3376817~~3375547 3377'957 3:~77'0~,7 33 77~+2 7 3415777:~431777 34333 77 3435737 3~36377 343~ 737 3437727 3455577" 3455 757 3457 3:~ 7 345 7~57 3457477 3457567 3457657 3457747 3463767 3465757 3466677 ~4~,5737 34~7337 34~,2557 3467557 3471577 34731 77 3473357 3/+73537 3473717' 3.473727 34 7~+ 7,':-, 7 ~'+ 25 177....3475477.....:~4 75 5~7........~475657.~ 347~. 747....3476727_.~ 3507 577 3507757....3612777 ~.... 35 13~77 3514777 3515377 3515~,77 3515737 3516757 3517277 3517477 3517557 3517657 0 3572777 3523877.. ~573767 35~.57'57..........~527177.......3527357 35~7537 3527.~57 3527717 3532577 t~ 35 327 37 3.932 7.~7 35? 3357 35 3X477 353363.7 35.33727 35~4757 3535 357 3535357 3536 337 35:~5657 35~7157 3577_~57 35375~7........ 35~.1777...........35z~3~77....... 3543757.............3545577 3547:337.......3547557....... 3.'51,7557 3~47~67 3551377 3552377 3552577:~553357 3553477 35536C7 355371 7 3554677 __ 35~4757........ ~55~ 5.~ 7........35~5637 _.~555~ ~ 7 3555557........35567 ] 7.......35571-67......3557257......35~,!377....3561757 35 ~27 37 ~563277 35~ 3337 3563357 3563537 3563717 3564 757 356'5 55 7 3565557 3566~77'~5~5~5 7 356:f/-5 7 3571277 ~571357 ~57~_357 35.72557 3572717 357447'7 357~557 3574717 3-375517 ~6!3377' 3513.737 3615377 3615757 3517337 3817557 3617637 3523677 3623757 3.*:'~') 77.7_............36~1677:.....36~ 17.~7..........3632577 3,53'~ 557.......:__,3.6 3~, 377.........3674737........._3835177 ~__ 357.~357. _3635557 3.~3~'71 7 3635477 3637157 ~645737' 36.47277 3647637 3653277 3653337 3555557 3656277 36_c. 7137 3657237 ~651377 366.)757 36~3777 3663z, 77 3663637...........36~577 _ 3664757........ 3665?77 _~.*, 5'3 337 36~,5~77 3.*-8~537 36 57! 37 3657437 3570737 3671337 3673 *)77 35 73437 36 74277 3675 237 367.5'23 7 3 7~2777 3703377 37.d. 3677.........370537-7 3706737.......3 707~.77........3707337......... 371~577 371~737 3723777 372267? 3723~.77 3723537 3726477. 37~1277 3731337 3732177 3733077 3734277......374>~.77.....37/+3] 77..........~745777..........375z+177...........3761177..................................... I

19TH DEGRE POLYNOMIALS, 17 Ni 4Z ERO COE-F FIC IENTS 2757777 2377377 2377577 2377775 2575777 2577577 2577677 2577773 2676777 2677775 2733777 2737r77 2737737 2737773 2756777 2757773 2765777 2767767 2771777 2773377 42773737 2775 k77 775737 2775767 2775775 2 776 5677 2776757 2777373 2777477 2777537 277 %7?77757' 777717 2777747 3137777 3177677 3275777 3277737 3317777 3335777 I 37A77 7 77 7 32 177 7 3357677 3357737 3363777 3 33577773 337 1777 33 73577 3373757_'75 137 7177 377357 3 77637 3377657 3473777 3477677 3533777 3537377 3557677 3557767 -r72c77 35 7377 3573677 3574777?-576577 3 576737 3577277 3637377 3637577 3)E51 77 7 7 77 3 57737 - Th66777 3667577 3673377 3675737 3675757 35757577 3677337 3677477 371577 ~3716777 3717737 3727377 3727737 3731777 3736377 3736577 3737177 3 7 /4ti7 7 7 74?Go 7 7 7 374 7 7 7 37597777. 376 1377 T__ _ _ _E ARE NO 11TH DEGREE POLYNOMIALS WITH 19 NONZERO COEFFICIENTS 72 7 —~ --------- --- -- --- ----------------------- - -----— ~- -- --- ---- --- ~ —~ — -- _ —-~ --— ~3 ~ ~ ___~__ ~ ~_~ _ ___ _ _ __ _____ ~ _~_~ _ ~ ~, - -, -— ~ —.-~ I ---- --- - ------ --- --- ---- -

RE FERENCES 1. J. J. SStiffler, Space Technology V5: Telecomn-municationls NAFA FP- 69, 1966. 2. C. C. Hoopes, Pseudo-Random Number Generators Using Linear Recurrence Relations, Technical Report No. 193, Cooley Electronics Laboratory, The University of Michigan, Ann Arbor, Michigan, July 1968. 3. G. C. Anderson, et al., "Pseudo-Random and Random Test Signals," Hewlett-Packard Journal, September 1967. 4. J. C. Steinberg and T. G. Birdsall, "Underwater Sound Propagation in the Straits of Florida, " Jour. Acous. Soc. Am., Vol. 39, NTo. 2, February 1966, pp. 301-315. 5. T. G. Birdsall and M. P. Ristenbatt, Introduction to Linear Shift-Register Generated Sequences, Technical Report No. 90, Cooley Electronics Laboratory, The University of Michigan, Ann Arbor, Michigan, October 1958. (AD255-380) 6. B. Elspas, "The Theory of Autonomous Linear Sequential Networks, " IRE Trans. on Circuit Theory, Vol. CT-6, No. 1, March 1959. 7. W. H. Kautz, Linear Sequential Switching Circuits, HoldenDay, San Francisco, California, 1965. 8. W. W. Peterson, Error Correcting Codes, MIT Press, Cambridge, Massachusetts, 1961. 9. S. W. Golomb, Shift Register Sequences, Holden-Day, San Francisco, California, 1967. 104

105 REFERENCES (Cont. ) 10. C. C. Hoopes and R. N. Randall, Study of Linear Sequence Generators; Technical Report No. 165, Cooley Electronics Laboratory, The University of Michigan, Ann Arbor, Michigan, June 1966. 11. E. J. Watson, "Primitive Polynomials (Mod 2)," Mathematics of Computation, Vol. 16, No. 79, July 1962. 12. R. W. Marsh, Table of Irreducible Polynomials over GF(2) through Degree 19, October 24, 1957.

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DOCUMEHNT CONtROL DATA. R & D (.rrltrIty rls * lefltel#lon nf tft*l, nroy cf h flrrf nrid Ind rtin onnrnoltilnn n.la Atr r iantrred when lhe overall rpotf o I cIlenN lled) I. gOItNA tING AC tIV ITV (Cotporft caliuthr) 2a. REPORT SECUnPTY CLASSIFICATION Cooley Electronics Laboratory Unclassified The University of Michigan 2h.. GRUP Ann Arbor Michigan 48105.. nCPORT trITLE An Ordered Table of Primitive Polynomials over GF(2) of Degrees 2 through 19 for Use with Linear Maximal Sequence Generators 4. otIEs rI P tIV o t (r'pe of r-poFt tatd IncltslIve datuO) Technical Memorandum No. 107 - July 1972: AU" THO-S (FIrtat name, middle Int-al, Iaa na..me) K. Metzger, Jr. and R. J. Bouwens 4. REPORT DATIE 7A. tOTAL NO. OF PAGES lb. NO. OF REFS July 1972 120 12 s4. CONTRAC t OR GRANT NO. Oa. ORIGINATOR'S REPORT NUMBERISI N00014- 67-A- 0181-003 5 004860- 2- M b. PROJECT NO. C. eb. OTHER REPORT NO(s) (Any other numbers that may be aessegned thli report) TM 107.., O1. DISTRIBUTION STATEMENT Approved for public release; distribution unlimited. ii. SUPPLEME TARY NOTES 2I. SPONSORING MILITARY ACTIVITY Office of Naval Research Department of the Navy Arlington, Va. 22217 13. AbStRACT This memorandum contains a brief discussion of the practical aspects of generating binary linear maximal sequences. Included is a table of all primitive polynomials over GF(2) of degrees 2 through 19, not including reverses. These polynomials are ordered by the number of nonzero coefficients as well as by degree. This table is felt to complement other available tables. D OR 1473 DD I.o,,.... - *:..... *..;!:, _.,- _,

Security Classification I 4. LINK A LINK S LINA C KEY WORDi.....:.5.-.. ROLE WT MOL W "'OLE^, WT Primitive polynomials Linear maximal sequences Sequence generators Binary sequences Table UciIP ==~-wifcl~

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