THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Electrical Engineering Information Systems Laboratory Technical Note ON THE NUMBER OF CLASSES OF (n,k) SWITCHING NETWORKS Michael A. Harrison 0ORA Project 04879 under contract with: UNITED STATES AIR FORCE AERONAUTICAL SYSTEMS DIVISION CONTRACT NO. AF 33(657)-7811 WRIGHT-PATTERSON AIR FORCE BASE, OHIO administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR December 1962

TABLE OF CONTENTS Page SUMMARY v I. INTRODUCTION 1 II. THE TOTAL NUMBER OF CLASSES 5 III. SYMMETRIES IN THE RANGE 7 IV. COMPLEMENTATION IN THE RANGE 14 V. LINEAR GROUPS IN THE RANGE 26 VI. MISCELLANEOUS RESULTS AND COMMENTS 42 REFERENCES 43 iii

SUMMARY An (n,k) switching network is defined as an n-inputk-output network such that associated with each output is a Boolean transmission function of the n inputs. If we allow a group (Y on the inputs and a group 4 on the outputs, then the family of networks is decomposed into equivalence classes. In this paper the numer of equivalene classes is derived for the important groups encountered in switching theory. v

I. INTRODUCTION Many writers have considered the problem of classifying Boolean functions under various groups (cf. 1, 4, 10, 14) In this paper we shall show how to extend the approach that this writer has taken in References 4, 5, 6, 7, in order to count the number of classes of sequences of k Boolean functions of n variables. We shall be initially interested in classifying sequences of k Boolean functions of n variables under some transformation groups on the n variables. One may think of the functions in these sequences as the transmission functions of a switching circuit having k outputs. See Fig. 1 for a diagram of the generic network which we shall call an (n,k) network. Our (n,k) networks realize the (k,n) sequences of Povarov12 as their behavior. Xn fl(xl,...,xn) f2(xl,...Yxn) xn ~,-xn) Fig. 1. One may take either of two points of view in this investigation. Networks can be characterized by (1) their structure, i.e., the placement of certain components in arbitrary arrays or (2) by their behavior, i.e., the 1

terminal relations. The former problem has been handled very neatly by Ninomiyall for the group of complementations and permutations. We shall not extend his results here, but we shall take the behavioral point of view. Some related results of a special nature have been obtained by Sagalovitch13 who obtained the number of one input k output networks whose transmission functions are non-trivial Boolean functions of n variables. In this paper, the "behavioral" aspect of the problem is completely solved for all the groups commonly studied in switching theory. Dramatis Personae.n n 1. ) is the group of all 2n complementations of variables. This r2 group was first studied as a group on Boolean functions by Ashenhurst.1 2. 3 denotes the symmetric group on the n variables. The order of n?' is n!; this group has been studied in Reference 4., n 3. n. is the smallest group containing and. It has been 0 n 2 n studied by a great many people; References 1, 4, 10, 14 are a small subset of the entire class of papers. 4. GLn(Z2) is the general linear group on the variables; the group has 14 7 been studied by Slepian and Harrison.7 n 5. ( (Z2) denotes the least group containing g and GLn(Z2); this n 2 group is the affine group on the variables and has been studied by Nechiporuk and Harrison.7 2

Before proceeding to our method, we shall briefly review the famous theorem of Plya which will be used. We shall use a form of the theorem adopted from De Bruijn.2 Let F be the class of all functions from a finite set D to a finite set R. Suppose D has s elements and that lJ is a permutation group of degree s and order g acting on D. Two functions fl, f2 C F are called equivalent if there exists a permutation a e 4/ such that fl(d) = f2(o(d)) for all d E D(a(d) denotes the image of d under the permutation a). Consider R to be represented r as the union of r disjoint subsets, i.e., R = (J Ri and Ri f Rj = if i i j. i=l Let kl,... kr be a partition of s. Polya's theorem tells us the number of equivalence classes of functions from D to R such that for ki values of d e D, the image f(d) e Ri for i = l,...,r. To every set Ri, we attach an indeterminate xi and define'i to be the number of elements in Ri for i = 1,...,r. The figure counting series is defined as r (X1,'...,Xr) = j Vixi i=l Usually the convention is adopted of taking x1 = 1. Let P(xl,...,xr) be the multi-variate generating function of the numbers that we are seeking, that is, the coefficient of x1... xr is the number of classes of functions with the property that for ki values of d e D, f(d) e Ri where i = l,...,r. P(xl1,.,xr) is often called the configuration counting series. Before stating Polya s theorem, we must develop the concept of the cycle index polynomial of f (Zyklenzeiger), denoted by ZV Let fl,...,fs be s 3

indeterminates, and let gjlj2.-s be the number of permutations of having ji cycles of length i for i = 1,2,...,s, so that S ii = s (1) i=l Then we define 1 ~ flj l f aj = g X flgJl,J2... J2 fs 9 M (J) where the sum is taken over all partitions of s which satisfy (1). Now we can finally state Polya's theorem which reduces the problem of determining the number of equivalence classes to the determination of the figure counting series and the cycle index polynomial. Theorem 2.1 (Polya). The configuration counting series is obtained by substituting the figure counting series into the cycle index polynomial of 6/. Symbolic ally P(xl,...,xr) = Z (R(xi,...,xxr), f(l... ),...,(. or ) / In our applications to single Boolean functions, Polya's theorem takes an even simpler form. Since Boolean functions are mappings from {O,11n into (0,1), we find that D = O,ln, s = 2n, and R = (0,1) with r = 2. Taking R1 = (0) and R2 = ({1 and associating the indeterminate 1 with R1 and indeterminate x with R2, the figure counting series becomes r(x) = 1l+x Before proceeding to the theoretical results, we give an example of two 4

networks to be considered equivalent under. Using the notation of Reference 4, let a = (2,3) e ~3be applied to the inputs of the top network of Fig. 2 to give the bottom network of the figure. X1 --- ------- fl1(X1,X2,X3) = X1X2X3 ~X2 ------- f2(xl,x2,x3) = Xl+X3 X3 ---- ----- f3(Xlx2,X3) = X1+X3 f4(Xl,x2,x3) = 1 X1 A —-=- -- -------- - fi(Xl,2,X3) = X1X2X3 f2(Xl,x2,X3) = X1+X2 f3(X1,X2,X3) = X1+X2 AX, —-—' 3- ------ f4(X1,X2aX3) = 1 Fig. 2. II. THE TOTAL NUMBER OF CLASSES Our problem is now to count the number of classes of (n,k) networks under any group f acting on the domain of the functions, {0,1}n The total number of such networks is 2k2 since thereare 2 choices for each one of the k outputs. In order to modify Polya's formula for counting the number of classes of networks, we note that no change has been made in the way that I acts on (0,1n. The only change is in the range. We are now looking at mappings n k from {0,1} into (0,1} so that we need a new figure counting series. Obviously 5

2k (x1,...,xk) = A xi i=l Usually xi is chosen to be one, but this is of no consequence. Theorem 1. The number of equivalence classes of (nk) networks (sequences of k Boolean functions of n variables) under a group % is given by z (2k,...2 k) Proof. To count the total number of classes, one takes xi = 1 for i = 1,...2k in the figure counting series. Thus fi is replaced by 2k 1 i = 2k for i = 1,.,2k j=l Corollary 2. The total number of classes of (nk) networks under: is 1 2k2n k2n-l (2 +(2n 1)2k2 2n Proof. This follows from the fact that, = 1( n +(2n 1)f 2 ) 2n (fl + (2n 2 1)f2 2 Cf. References 1 and 4. It is interesting to note that the class of all mappings from (0,l1n into {0,1O k forms a Boolean algebra of 2k2 functions in the natural way. It appears that the generalization of many problems in switching theory to the multiple output case can be handled naturally from this point of view. This observation has already been verified for the multiple output minimization problem. 6

The calculations have been carried out for the five groups under discussion and the results are given below in Tables 1 through 5. Certain facts may be deduced from the tables. For examplethe number of classes under ~r and GL1(Z2) is 4k since these groups consist of the identity alone. III. SYMMETRIES IN THE RANGE While the results of the previous section are of some interest, certain networks are considered non-equivalent which differ only in the order of the outputs. We wish to enlarge our definition of equivalence of networks by allowing permutations in the range. More precisely, let the symmetric group on k letters, k, act on the range, and consider two networks equivalent if and only if there is an a! e C and a permutation a E ^t such that fl(d),...,fk(d)) = (g(l) (Cd),..g(.)(Od)) for every d e {O,1}n This instance is a special case of the following theorem of De Bruijn.2 Theorem 3. (De Bruijn). If 0 is a group on the domain D of a family of functions, and. is a group on the range R of the function (D = s) and R = r), then the number of equivalence classes of functions is given by l,^ — )Z h (hi,... hr) evaluated at zl = zs =... = zs = 0 where ht = exp t Zkt for t = 1,...,r k=l In order to carry out calculations with this theorem, the following lemma was proved in Reference 5. 7

TABLE 1 ~n THE NIUMBER OF CLASSES UNDER [2 n k=l k=2 k=3 k=4 1 3 10 36 136 oo 2 7 76 1,072 16,576 3 46 8,416 2,100,736 536,928,256 4 4,336 268,496,896 17,592,201,773,056 1,152,921,508,633,378,816 --,, _,, _ i.....,

TABLE 2 THE NUMBER OF CLASSES UNDER \ n k=l k=2 k=3 k=4 1 4 16 64 256 \o 2 12 160 2,304 34,816 3 80 13,056 2,928,640 724,238,336 4 3,984 1,833,052,160 11,745,443,774,464 768,684,844,023,545,856

TABLE 3 THE NUMBER OF CLASSES UNDER Fi n k=l k=2 k=3 k=4 1 3 10 36 136 o 2 6 55 666 9,316 3 22 1,996 384,112 91,604,416 4 402 11,756,666 735,192,450,952 48,047,227,408,513,056

TABLE 4 THE NUMBER OF CLASSES UNDER GLn(Z2) n k=l k=2 k=3 k=4 1 4 16 64 256 H 2 8 80 960 13,056 3 20 1,056 135,040 27,700,736 4 92 320,416 14,333,211,520 916,495,047,958,016

TABLE 5 THE NUMBER OF CLASSES UNDER'n(Z2) n k=l k=2 k=3 k=4 1 3 10 36 136 H 2 5 35 330 3,876 3 32 271 22,060 3,741,616 4 382 29,821 920,737,780 57,374,820,122,576, —------------—, —J

Lemma 4. A term hil... hrJr in Z gives rise to zig (E ^tJtn X tt) tll tls The only information required before our calculations can begin is the determination of the cycle index of k. Since there are 2k elements of (0,l}1, we need a representation o f degree 2. Fortunately, such a representation was constructed in Reference 4. The result is k' Ji (7-1 kxj i k! Xk k (fd e(d). i! () bdli where the sum is over all solutions of k iji = k i=l and e(d) = d j 2tl (~). The definition of the cross operation (X) is given t d in Reference 4, and I(a) denotes the Mobius function. For the sake of reference, the results of De Bruijn's theorem are worked out below for 1 < k < 4. Theorem 5. The number of classes of (nk) networks with a group on the domain and the symmetric group on the range is given below for 1 < k < 4. k=l k = 1 Z(...) k=2 2 (Z ( ) +Z (,...) k = (Z,...)+ j5z(4,,...) + 2Z(22,8,...)) k = 4 2 4 (z(1l6,...) + 6z (-(8,16,... ) + 5z~(4,l6,... ) + 8Z%(4,4,16,...) + 6zo(2,4,2,16,... )) 13

where the notation Z (f,-... -,fm,. ) means fp = fq if and only if p - q mod m. The calculations have been carried out and are given in Tables 6 through 10. The number of classes with (say) 2 on the domain and S on the range is den noted by T ( [, k). IV. COMPLEMENTATION IN THE RANGE Suppose we allow ourselves to complement the k outputs of our networks and enlarge our definition of equivalence by considerirg two networks to be k equivalent if there isana e e and an i = (il,...- ik) E:, such that (fi(d),...,fk(d)) = (fl l(c(d)),...,fkk(ca(d)) for all d e (0,1) The notation fjij(d) is defined as i. fj(d) if ij = 0 fj J(d) = for j = l,...,k fj(d) if ij = 1 The number of such classes can again be determined from De Bruijn's theorem. Theorem 6. The number of classes of (n,k) networks with a group 6 on the k domain and the complementing group J on the range is 1, - ----- Xk 2' -- k (z(2k,...,2k) + (2k _ 1) (0,2,..., 2-k 1 Proof. The cycle index of is known to be Zk = 2k (f + (2 - 1)k2 ) 2 The result follows from Theorem 35 14

TABLE 6 T([, tA) n k=l k=2 k=5 k=4 1 3 7.13 22 \-f 2 7 46 237 1,056 3 46 4,336 356,026 22,921,696 4 4,336 134,281,216 2,932,175,712,336 48,042,795,872,587,776,,,,,....,,m i, ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~~ ~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

TABLE 7 T(n, ) n k=l k=2 k=3 k=4 1 4 10 20 35 a 2 12 88 484 2,180 3 80 6,616 497,760 31,017,356 4 3,984 916,539,904 1,957,701,217,238 32,031,538,353,966,080

TABLE 8 T(n, A") n k=l k=2 k=3 k=4 1 3 7 13 22 -- 2 6 32 158 652 3 22 1,056 66,336 3,945,992 4 402 5,884,954 122,543,247,874 2,002,161,498,159,934

TABLE 9 T(GLn(Z2), k) n k=l k=2 k=3 k=4 1 4 10 20 35 o 2 8 46 220 897 3 20 556 23,932 1,214,454 4 92 160,932 2,390,063,996 38,192,408,400,654

TABLE 10 n k=l k=2- k=3 k=4 1 3 7 13 22 \10 2 5 22 87 314 3 10 153 4,148 169,495 4 32 15,209 153,668,757 2,391,065,770,697

The special case when k = 1 has been previously investigated and is the topic of References 3, 5, and 10. The calculations are shown below in Tables 11 through 15, where T(, i ) denotes the number of equivalence classes 2 of (n,k) networks with a group on the domain and on the range. It is somewhat artificial to consider complementation of the outputs only. One would prefer to consider both symmetries and complementations on the range. This suggests considering the least group containing fk and ~k' j. This group, k k, is very well known and its cycle index is known to 2 k be ~ e(d) g(d) Zk k, k2 fd " + fd d12i where the sum is over all partitions of k, i.e., over all non-negative integer k Js k solutions ijj = k. e(k) 2 and i=l djk g(2k) 1 d/2 2k dtk d 2k where i(a) is the Mobius function. Using Lemma 4, the polynomials to be evaluated are written below. Theorem 7. The number of classes of (n,k) networks with a group f on the domain and the group of complementations and permutations on the range k -- - - -- --- is given below for 1 < k < 4. 20

TABLE 11 n k T([2 [2) n k=l k=2 k=3 k=4 1 2 4 8 16 H 2 5 22 176 1,216 5 30 2,128 265,728 33,611,776 4 2,288 67,127,296 2,199,025,221,832 72,057,598,064,459,776

TABLE 12 <k~ n k=l k=2 k=3 k=4 1 2 4 8 16 o 2 6 40 288 2,176 3 40 3,264 366,080 45,264,896 4 1,992 458,263,040 1,468,180,471,808 48,042,802,751,471,616

TABLE 13 T(an', [2 n k=l k=2 k=3 k=4 1 2 4 8 16 \, 2 4 19 106 676 3 14 556 49,008 5,742,016 4 222 2,945,738 91,901,007,752 2,877,952,247,834,656

TABLE 14 T Z ),.k n k=l k=2 k=3 k=4 1 2 4 8 16 -- 22 4 20 120 816 3 10 264 16,880 1,731,296 4 46 80,104 1,791,651,440 51,030,940,434,876

TABLE 15 n k=l k=2 k=3 k=4 1 2 4 8 16 ro \.1 2 3 11 50 276 3 6 79 2,908 236,176 4 18 7,621 115,125,476 3,585,934,560,176

k =1 2 (Z(2,...) + Zr(,...)) k = 2 7 (Zj(4,...) + 5Z6(,+4,...) + 2Z(2,4,...) + 2Z (0,0,0,4,...)) 1 - k = 35 (zo(8,...) + 13Zo(0,8,...) + 8Zy(2,2,8,...) + 8Z (0,2,o,2,0,8,...) + 6Zy(T7,...) + 122(0,0,0,8...)) k- 4 (z(i6,...) + 51Z(0,1,....) + 48zf(2,4,2,16,...) + 48ZZ (0,0,0,0,0,0,0,16,...) + 12ZV(7I,...) + 84z (0,o,o,l6,...) + 12Z(4,16,...) + 32Z)(4,4,l6,..) + 96zj(o,4,o,4,o,l6,...)) Using this result, the number of classes has been computed and is tabulated below in Tables 16 through 20. V. LINEAR GROUPS IN THE RANGE Theorem 3 and Lemma 4 allow us to consider any group 7'on the domain and any group on the range. We now complete our analysis of all remaining cases by allowing GLn(Z2) and n(Z2) on the range. Because of the length and complexity of the cycle indices of these groups, we shall not exhibit the explicit formulae to be used in the calculations; instead the reader is referred to Reference 7. The results are given below in Tables 21 through 30. 26

TABLE 16 n k=T. k=2 k3 k k) n k=l k=2 k=3 k=4 1 2 3 4 5 - 2 5 18 51 122 3 30 1,200 45,777 1,476,032 4 2,288 33,601,536 366,543,984,720 3,002,400,587,673,600

TABLE 17 T(b an k) ~n k=l k=2 k=3 k=4 1 2 3 4 mC 2 6 24 98 194 3 40 1,676 63,440 1,992,430 4 1,992 235,384,592 244,644,311,176 2,002,158,848,764,301

TABLE 18 T(On, k) n k=l k=2 k=5 k=4 1 2 3 4 5'o 2 4 13 34 640 3 14 308 8,906 257,658 4 222 1,476,218 15,320,103,918 125,147,156,711,032

TABLE 19 T(GLn(z2), k) n k=l k=2 k=3 k=4 1 2 3 4 5 o ~2 4 13 36 95 3 10 146 3,178 80,036 4 46 40,422 298,842,656 2,387,346,322,704

TABLE 20 T(. n(Z2) Ik) n k=l k=2 k=3 k=4 1 2 3 4 5 p2 3 8 19 41 3 6 49 632 11,917 4 18 3,963 19,245,637 149,471,180,139

TABLE 21 T([2, GLk(Z2) n' k=l k=2 k=3 k=4 1 3 4 4 4 ia\j ^2 7 21 27 28 3 46 1,531 14,056 39,839 4 4,336 44,782,251 104,751,025,086 57,280,291,273,345

TABLE 22 T( 2 GLk(Z2) n k=l k=2 k=3 k=4 1 4 5 5 5 2 12 35 46 47 3 80 2,266 19,930 55,767 4 3,984 305,552,546 69,945,183,326 38,191,772,055,973

TABLE 23 T(6<n, GLk(Z2)) n k=l k=2 k=3 k=4 1 3 4 4 4 -- 2 6 16 21 22 3 22 392 2,920 7,923 4 402 1,966,074 4,379,140,552 2,387,316,975,717

TABLE 24 T(GLn(Z2), GLk(Z2)) n k=l k=2 k=3 k=4 1 4 5 5 5 \-fl 2 8 20 27 28 5 20 206 1,204 3,071 4 92 54,155 85,552,416 45,568,388,658

TABLE 25 T(.Yn(Z2), GLk(Z2)) n k=l k=2 k=3 k=4 1 3 4 4 4 2 5 11 15 16 3 10 64 276 653 4 32 5,276 5,534,421 178,471,391

TABLE 26 T([2 rtk( 2,)) n k=l k=2 k=3 k=4 1 2 2 2 2 -5 2 5 9 10 10 3 30 443 2,104 3,763 4 2,288 11,211,435 13,098,898,366 3,585,768,962,689

TABLE 27 T(~n', &k(Z2)) n k=l. k=2 k=3 k=4 1 2 2 2 2 o 2 6 11 12 12 3 40 590 2,828 5,066 4 1,992 76,384,114 8,747,130,342 2,390,901,609,645

TABLE 28 T(v, n' tk(Z2)) n k=l k=2 k=3 k=4 1 2 2 2 2 \ 2 4 7 8 8 5 14 124 504 88o 4 222 494,298 547,849,868 539,100,216

TABLE 29 T(GL)(Z), k(Z2)) n k=l k=2 k=5 k=4 1 2 2 2 2 0 2 4 7 8 8 3 10 60 214 568 4 46 153,75533 10,724,116 2,854,852,585

TABLE 30 T(n1(Z2), atk(Z2)) n k=l k=2k=5 k=4 1 2 2 2 2 H 2 56 6 3 6 24 70 116 4 18 1,430 700,175 179,125,249

VI. MISCELLANEOUS RESULTS AND COMMENTS One might be interested in comparing our results with some related results on invertible functions. In both cases we consider n-input,n-output networks with groups on the domain and the range, but in References 6 and 8 the functions are required to have inverses. Comparing the results of these references against T(<, X) for (n,n) networks, one finds that the ratio of the number of classes of invertible functions to T( /, I) approaches zero for large n. Thus the invertible classes are comparatively rare. One degenerate group has not been discussed, namely the identity group on the n variables, k. n has order 1, degree 2n, and its cycle index is Our results with on the range reduce to Z(22) as one is fl ~ Our results with o on the range reduce to Z.(2k 2k) as one would expect. Separate calculations are required if n is applied to the n domain and an arbitrary group ~ is used on the range. These calculations will not be performed here because of the limited interest in the results and the size of the numbers. It is sometimes of interest to have lower bounds on the number of classes. The following theorem gives the desired bounds. Theorem 8. A lower bound on the number of classes of (n,k) networks with a group of order g on the inputs and a group of order h on the outputs is given by n h Zy(2 g.O.2 ) > 2 1 Z (2k 2k ) > 1 2k2n Proof. The argument simply takes the largest terms in the polynomials given in Theorem 3. Note that for k = 1, and 0 = 11, the bound reduces to - 22 which is a well known7'8 lower bound for the number of classes of g Boolean functions with a group of order g on the domain. 42

REFERENCES 1. Ashenhurst, R. L., "The application of counting techniques," Proceedings of the. Association of Computing Machinery, Pittsburgh Meeting (1952), pp. 293-305. 2. De Bruijn, N. G., "Generalization of Polya s fundamental theorem in enumerative combinatorial analysis," Koninklijke Nederlandse Akademie Von Wetenschappen, Series A, Vol. LXII, No. 2 (1959), pp. 56-59. 3. Elspas, B., "Self-complementary symmetry types of Boolean functions," IRE PGEC, Vol. EC-9, No. 2, June (1960), pp. 264-266. 4. Harrison, M. A., The Number of Transitivity Sets of Boolean Functions, The University of Michigan Technical Note 04879-3-T, June (1962). 5. Harrison, M. A., The Number of Equivalence Classes of Boolean Functions Under Groups Containing Negation, The University of Michigan Technical Note 04879-4-T, June (1962). 6. Harrison, M. A., The Number of Classes of Invertible Boolean Functions, The University of Michigan Technical Note 04879-5-T, July (1962).. Harrison, M. A., On the Classification of Boolean Functions by the General Linear and Affine Groups, The University of Michigan Technical Note 04879-7-T, September (1962). 8. Lorens, C. S., Invertible Boolean Functions, Space General Corporation Report, July (1962). 9. Nechiporuk, E. I., "On the synthesis of networks using linear transformations," Doklady Akad Nauk 123:610-612, No. 4, December (1958): available in English in Automation Express, April (1959), pp. 12-13. 10. Ninomiya, I., "On the number of genera of Booleanfunctions of n variables, Memoirs of Faculty of Engineering of Nagoya University, Vol. 11, No. 1-2, November (1959), pp. 54-58. 11. Ninomiya, I., "On the number of types of symmetric Boolean output matrices' Memoirs of Faculty of Engineering of Nagoya University, Vol. 7, November (1955), pp. 115-124. 12. Povarov, G. N., "A mathematical theory for the synthesis of contact networks with one input and k outputs," Proceedings of an International Symposium on the Theory of Switching, Part II, Harvard University Press, Cambridge, Mass. (1959), pp. 74-94. 45

13. Sagalovich, I. U., "On the number of types of symmetry for contact (l,k) -pole networks," Doklady Akad Nauk 130:72-73, No. 1, January (1960); available in English in Automation Express, March (1960), pp. 39-40. 14. Slepian, D., "Some further theory of group codes," The Bell System Technical Journal, Vol. XXXIX, No. 5, September (1960), pp. 1219-1252. 44

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