We extend the results of our previous paper [8] to the nonlinear case: The Lloyd polynomial of the covering has at least R distinct roots among 1, ... , n, where R is the covering radius. We investigate PWC with diameter 1, finding a partial characterization.  We complete an investigation begun in [8] on linear PMC with distance 1 and diameter 2

Cohen, G. D.

Litsyn, S. N.

Mattson, H. F., Jr

English

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Electrical Engineering and Computer Science - Technical Reports College of Engineering and Computer Science 9-1991 On Perfect Weighted Coverings with Small Radius G. D. Cohen S. N. Litsyn H. F. Mattson Jr Follow this and additional works at: https://surface.syr.edu/eecs_techreports  Part of the Computer Sciences Commons Recommended Citation Cohen, G. D.; Litsyn, S. N.; and Mattson, H. F. Jr, "On Perfect Weighted Coverings with Small Radius" (1991). Electrical Engineering and Computer Science - Technical Reports. 139. https://surface.syr.edu/eecs_techreports/139 This Report is brought to you for free and open access by the College of Engineering and Computer Science at SURFACE. It has been accepted for inclusion in Electrical Engineering and Computer Science - Technical Reports by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. SU-CIS-91-35 On Perfect Weighted Coverings with Small Radius G. D. Cohen, S. N. Litsyn, and H.F. Mattson, Jr. September 1991 School of Computer and Information Science Syracuse University Suite 4-116, Center for Science and Technology Syracuse, New York 13244-4100 ON PERFECT WEIGHTED COVERINGS WITH SMALL RADIUS 1 Gerard D. Cohen Ecole N ationale Superieure des Telecommunications 46 rue Barrault, C-220-5 75634 Paris cedex 13, France Email: Cohen@inf.enst.fr Simon N. Litsyn Dept. of Electrical Engineering-Systems Tel-Aviv University Ramat-Aviv 69978, Israel Email: litsyn@genius.tau.ac.il H. F. Mattson, Jr. School of Computer and Information Science 4-116 Center for Science & Technology Syracuse, New York 13244-4100 Email: jen@SUVM.acs.syr.edu, jen@SUVM.bitnet Abstract: We extend the results of our previous paper [8] to the nonlinear case: The Lloyd polynomial of the covering has at least R distinct roots among 1, ... , n, where R is the covering radius. We investigate PWCwith diameter 1, finding a partial character-ization. We complete an investigation begun in [8] on linear PMC with distance 1 and diameter 2. 1This paper was presented at the French-Soviet Workshop in Algebraic Coding, ENST, Paris, 22-24 July 1991. It will appear in the Proceedings, to be published by Springer in the LNCS series. 1 Introduction Much attention has been devoted to the problem of classifying perfect codes (See [13, 15]). Further generalizations of perfectness were introduced in [10, 2, 11, 14]. For all these codes the diameter of the covering spheres equals the covering radius of the code which by use of Delsarte's results leads to a very rigid set of possible parameters. This framework was broadened by introducing new types of perfect configurations [5, 6, 12, 16]. All these extensions fall under the concept of perfect weighted coverings (PWC) first considered in [8). Although general, these definitions leave hope for a complete classification, at least for small diameter. The linear case with diameter at most 2 was considered in [8), where some motivation related to list decoding was given. We are pleased to acknowledge that this problem arose in discussions with I. Honkala in Veldhoven in June, 1990. 2 Notations and known results We denote by Fn the vector space of binary n-tuples, by d(·, ·) the Hamming distance, by C ( n, K, d)R a code C with length n, size K, minimum distance d = d( C) and covering radius R [9], [7). When C is linear, we write C[n, k, d)R, where k is the binary log of K. We denote the Hamming weight of x E Fn by lxl. For X eFn' A( X) = ( Ao( X)' Al (X) ... An (X)) will stand for the distance distribution of C with respect to x; thus Ai(x) := l{c E C: d(c,x) = i}l. For any (n + 1)-tuple M = (mo, m11 •.• , mn) of weights, i.e., rational numbers, we define the M- density of C at x as n (2.1) O(x) := L: mi ~(x) = <M,A(x)>. i=O We consider only coverings, i.e., codes C such that O(x) 2:::: 1 for all x. (2.2) C is a perfect M -covering if 0( x) = 1 for all x. We define the diameter of an M -covering as 8 := max{i: mi =f. 0}. To avoid trivial cases, we usually assume that mi = 0 fori~ n/2, i.e., 8 < n/2. Here are the known special cases. (2.3) Classical perfect codes: mi = 1 for i = 0, 1, ... 8. (2.4) Perfect multiple coverings (PMC): mi = 1/j fori= 0, 1, ... 8 where j is a positive integer. See [16] and [5]. (2.5) Perfect L-codes: mi = 1 fori E L ~ {0, 1, ... Ln/2J}. See [12] and [6]. (2.6) Strongly uniformly packed codes: mi = 1 fori= 0, 1, ... , e- 1 me= me+l = 1/r for some integer r. See [14]. (2.7) Uniformly packed codes (2, 11]. For these codes 8(M) = R(C), and the mi's are uniquely determined. The following necessary and sufficient condition was already in [8] in the linear case. For a perfect M-covering Cone gets from the definition: n L mi Ai(x) = 1 for all x. i=O Summing over all x in Fn and permuting sums, we get n L mi L Ai(x) = 2n. i=O xEF" Fori= 0, the second sum is ICI = K, fori= 1 it is Kn, and so on. For the converse we use the condition O(x) ~ 1. Hence we get the following analog of the Hamming condition. Proposition 2.1 A covering C is a perfect M -covering if and only if (2.8) 0 3 A Lloyd theorem In this section we prove Theorem 3.1 Let C be a perfect weighted covering with M = (m0 , m17 ••• , m 5). Then the Lloyd polynomial of this covering, L(x) := L mi Pn,i(x) 0:5i:55 has at least R distinct integral roots among 1, 2, ... , n. Proof. (Adapted from [1], Chapter II, Section 1, which records A. M. Gleason's proof of the classical Lloyd theorem.) The first part of the proof is identical to that of [8, Thm. 4.1]. We use the group algebra A of all formal polynomials with "'a. E Q, the field of rational numbers. Define (3.1) s := 2: mi 2: xa.. O~i~6 la.l=i We let the symbol C for our code also stand for the corresponding element in A, namely, (3.2) Then we find that (3.3) Characters on Fn are group homomorphisms of (Fn,+) into {1,-1}, the group of order 2 in qx. All characters have the form Xu for u E Fn, where Xu is defined as Xu(v) = (-1)u·v for u,v E Fn. We use linearity to extend Xu to a linear functional defined on A: For allY E A if Y = Ea.eF" "fa.Xa., then Xu(Y) := E"'a.Xu(a). It follows that Xu(YZ) = Xu(Y)xu(Z) for allY, Z EA. It is known [1, 9] that for any u E Fn, if lui = w, then (3.4) It follows that (3.5) Xu (E xa.) = Pn,i(w). la.l=i Xu(S) - L(w). From (3.3), furthermore, we see that Xu(SC) = Xu(S)xu(C) - 0 for all u =f:. 0. Let uo, Ut, ••• , UR be translate-leaders for C such that lui I = i. Define ci :=xu· c. Then (3.6) Define the symmetric subring A of A as the set of all elements Y of A in which the coefficient of xa depends only on the weight of a: (3. 7) y = L /a xa E A iff Va, bE F\ Ia I = lbl -+ /a = /b· aeFn The mapping T : A ~ A defined by 1 T(Y) := I L cp(Y), n. cp where cp runs over all n! permutations of the n coordinates of pn, maps A onto A. Furthermore, as the reader may easily verify, (3.8) VY E A, VZ E A, T(YZ) = YT(Z). Define Ci := T(Ci)· Applying (3.8) to (3.6), we see that SCi= Fn since, of course, S E A. Define also (3.9) K := { Z; Z E A, S Z = 0}. Thus K is the kernel of the linear mapping from A to A defined by Y ~ SY for all YEA. It follows from (3.8) that for any character Xu. such that Xu.(S) -=/= 0, VZ E K, Xu.(Z) = 0. Since A has dimension n + 1, its space of linear functionals also has dimension n + 1. Since every linear functional on A can be extended to one on A, then+ 1linear functionals on A obtained by restricting the Xu. to A, as Xu.h =: Xw for lui = w w = 0,1, ... ,n, are linearly independent. Suppose that pis the exact number of values of w E {0, 1, ... , n} for which Xw(S) -=/= 0. Since Xw(S)xw(K) = 0 for all w, it follows that Xw(K) = 0 for p values of w. Since SCi= Fn fori= 0, 1, ... , R, we see that S ( C i - C 0 ) = 0 for i = 1, ... , R. The elements Ci- C0 are linearly independent because Ci contains elements of weight i but of no smaller weight. We find that R ~ dimq K < n + 1 - p, since K is included in the intersection of the t kernels of the Xw mentioned above. But n + 1 - p is the number of Xw's which vanish on S; therefore Xw(S) = 0 for at least R values of w. Notice now that Xw(S) - L mi Pn,i(w). O~i$6 This finishes the proof. 0 4 A construction Definition 4o1 Let C(n, K, d)R and C'(n', K', d')R' be two codes. Set { 0 ifxEC Xc( X) = 1 otherwise. We extend xc to a mapping x :F"'"'' -+F"'' by setting where the Xi 's are in F"', for 1 < i < n', and x = ( x1 , x2, . o ox,,) is their concatenation. We are now ready to define C ® C' as follows: C ® C' = { z E F"'"'': x(z) E C'} o Proposition 4o1 C®C' has length nn', minimum distance min{ d, d'} and covering radius RR'. The proof is immediate. 0 Proposition 4o2 Let x and x' be such that d(x,C) = R,d(x',C') = R'. Suppose that AR(x) and Ak,(x') are independent of x. Then for C ® C' the coefficient ARR'(z) is the same for any z such that d(z, C ® C') = RR' and one has 0 5 PWC with diameter 1 Let us denote such a PWCby (n,m0 ,m1). From (2.2), A1(x) = 1/m1 for any x not in C. Hence m1 = 1/p, where pis an integer. This means that every two noncodewords have the same number of codewords at distance 1. ForcE C, we get: m0 + A1(c)jp = 1, hence A1(c) = p (1- mo) is a constant independent of c. Since A1(c) is an integer, so is m0p. Now the Hamming analogue (2.9) gives K (pm0 + n) = p 2"', which implies (5.1) n = p' 2i- mop, with p' I p. The case m0 = 1/p corresponds to the PMC mentioned in (2.4); it is solved in [16] and [8]. Let us give a few general constructions. Proposition 5.1 If there exists a PWC C(n, mo, mt), then for any l ~ 0 there exists a PWC C'(n + l, m0 -lmt, mt)· Proof. Let us define C' as the set of vectors ( c, f) in Fn+l, where c E C and f EF1• Let A be the distance distribution for C1 and A' that for C'. There are two possibilities for an arbitrary (x, f) EFn+l: (a) x E C. Then A~((x,f)) = A1 (x) + l. Evidently A~((x,f)) = 1. (b) x rt C. Then A~((x, f))= 0 by construction and A~((x, f))= At(x). 0 Proposition 5.2 If there exists a PWC C(n, m0 , mt), then there exists a PWC C'(ns, mo, mtfs). Proof. Apply construction® (Def. 4.1) with outer code C(n, mo, mt) and inner code the [s, s- 1] parity code. 0 Proposition 5.3 If there exists a PWCC(n,m0,m1 ), then there exists a PWC C'(n, m 0 /i, m 1 /i), fori a positive integer. Proof. Take the union of i cyclic shifts of code C. 0 Let us now turn to the special case when mo = 1. Proposition 5.4 A PWC with 8 = m 0 = 1 exists for n = p(2i- 1), m1 = 1/p. It can be achieved by a linear code. 0 See [8] for a proof of this result. In contrast to the linear case, [8, Prop. 5.4], we cannot characterize PWC with 8 = m0 = 1 here. However, we have a partial characterization: Proposition 5.5 A PWC(n,1,2-q) exists if and only if for some i n = 2q(2i-1). Such a PWC can be achieved by a linear code. Proof. If m1 = 2-q, then p' = 2q',q' ~ q, and (5.1) gives n = 2q(2i+q'-q- 1). The converse stems from Proposition 5.4. 0 We would like to point out that for some parameters satisfying {5.1) there is no corresponding code. Consider the case m0 = 1, m1 = 1/3. Proposition 5.4 gives the sequence of lengths n = 3 · 2i- 3. The other possibility is n = 2i- 3. The first code in this sequence would be a PWC with n = 5 and K = 12. Let us show its nonexistence. Proposition 5.6 A (5, 1, 1/3) PWC does not exist. Proof. We may assume the code contains the zero vector. Furthermore, it does not contain vectors of weight 1, since the minimum distance is 2 for m0 = 1. Every vector of weight 1 has to be covered by exactly two codewords of weight 2. There are exactly 5 codewords of weight 2, because if we consider the matrix of all such codewords, we see that each column has sum 2 (by the "coverage" condition just mentioned). Let x be any vector in F5 of weight 3. Each "1" in x is covered by two codewords of weight 2. That makes six codewords of weight 2. By the pigeonhole principle, two are equal, say to c E C. Then x is at distance 1 from c. So the code does not contain vectors of weight 1 and 3, and we cannot cover vectors of weight 2. 0 6 Linear PMC with diameter 2 (mo = m1 = m2 1/j) The purpose of this section is to summarize and extend results from [8]. 6.1 The case s = 1 Proposition 6.1 [8] The only PMC with s -j = 2. 1, d - 2 is the [2, 1, 2] code with 0 We assume now that d is equal to 1. To set the stage, we repeat some material from [8]: We find that the only possibility for the check matrix is the t-fold repetition of g(Si) (generator matrix of a simplex code of length 2i - 1) with l zero-columns appended, yielding n = t(2i - 1) + l. It amounts to appending all possible tails of length l to codewords described in Proposition 5.2. It is easy to check that there are 2 kinds of covering equalities (namely, vectors coinciding with, or being at distance 1 from, codewords on the first t(2i- 1) coordinates): This implies (6.1) mo + lm1 + G) (2i - 1 )m2 + (~) m 2 = 1 tm1 + (2i-l - 1) t 2m 2 + tlm2 = 1. t 2 - t(2i + 1 + 2/) + ( 12 + l + 2) = 0 which has discriminant (6.2) We get a PMCiff D = x 2 has integer solutions. For example, the values i = 3, l = 3, t = 14 yield the PMC [101, 98] with j = 644. Of course, for i = t we get 81 + 1 = x 2 having all odd x as solutions. Now we can characterize the solutions of D = x 2 • We need the following result: Proposition 6.2 (2i+1 - 7) is a square mod 2i+2 • Proof. Proof by induction on i. If x is a solution for some i, i.e., for a E N, x 2 =a 2i+2 + 2i+l - 7, then for any f3 E N to be chosen later on, and i 2:: 3: (x + 2i+1 f3 + 2i)2 = x2 + 2i+2 (x,8 +a)+ 2i+lx + 2i+l - 7 + 22i(1 + 4,82 + 4,8) = 2i+2 (xf3 +a+ x;l) + 2i+2- 7 mod 2i+3. Since x is odd, we can certainly find f3 to make x/3 +a+ x;l even. Then x + 2i+1 ,8 + 2i is a solution for i + 1. For i ::::; 2, the proposition is easily checked. 0 The first proof of this proposition was given by I. Shparlinski during the present Workshop. Obviously, the congruence has 4 roots. Denoting by a the one which lies in [0,2i+l], they are Now direct calculations lead to the solution of (6.2), giving the possible l. Then t is derived from (6.1). Theorem 6.1 Linear PMC with m 0 = m 1 = m 2 = 1/ j, d = 1 exist only for the following sets of parameters: 1 = (1222i+2 ± 2i+2/a + a2 _ 2i+l + 7 _ 22i)/2i+2 t = (2i + 1 + 21 ± J(2i + 1)2 + 2i+2l- 8)/2 n = t(2i - 1) + l k = n -i j = (2i-l- 1)t2 + t(1 + 1), for 1 E Z, provided lEN. 0 6.2 The case s = 2 We have found the following PMC codes C in this case (d = s = h = 2); see [8] for constructions. c [5, 1; 5] [5, 2, 2] [5, 3, 2] [10, 7, 2] (37, 32, 2] (8282, 8269, 2] j=1 j=2 j=4 j=7 j = 22 j = 4187 c.L [5,4; 2, 4] [5, 3; 2, 4] [5, 2; 2, 4] [10, 3; 4, 7] (37, 5; 16, 22] [8282, 13; 4096, 4187] The first is a classical perfect code. The notation [n, k; Wt, w 2 , .. . ] stands for an [n, k] code in which all nonzero weights are among Wt, w2 , ... . In the above codes C.L, since s = 2, both weights are present. All the above codes Care PMC codes. Conjecture 6.1 We conjecture the nonexistence of PMC with d = s = h = 2 other than those in the table. References [1] E. F. Assmus, Jr., H. F. Mattson, Jr., R. Turyn, "Cyclic codes," Final Report, Contract no. AF 19(604)-8516, AFCRL, April28, 1966. Sylvania Applied Research Laboratory, Waltham, Mass. (Document no. AFCRL-66-348.) [2] L. A. Bassalygo, G. V. Zaitsev, V. A. Zinoviev, "On Uniformly Packed Codes," Problems of Inform. Transmission, vol. 10, no. 1, 1974, pp. 9-14. [3] L.A. Bassalygo, G. V. Zaitsev, V. A. Zinoviev, "Note on Uniformly Packed Codes," Problems of Inform. Transmission, vol. 13, no. 3, 1977, pp. 22-25. [4] A. R. Calderbank, J. M. Goethals, "On a Pair of Dual Subschemes of the Hamming Scheme Mn(q)," Europ. J. Combinatorics, 6, 1985, pp. 133-147. [5] R. F. Clayton, "Multiple Packings and Coverings in Algebraic Coding Theory," Thesis, Univ. of California, Los Angeles, 1987. (6] G. D. Cohen, P. Frankl, "On tilings of the binary vector space," Discrete Math., 31, 1980, pp.271-277. [7] G. D. Cohen, M. G. Karpovsky, H. F. Mattson, Jr., J. R. Schatz, "Covering Radius-Survey and Recent Results," IEEE Trans. Inform. Theory IT-31, 1985, pp. 328-343. [8] G. D. Cohen, S. N. Litsyn, H. F. Mattson, Jr., "Binary Perfect Weighted Cover-ings," Sequences '91, Positano, 17-22 June 1991, Springer- Verlag Lecture Notes in Computer Science. To appear. [9] P. Delsarte, "Four Fundamental Parameters of a Code and Their Combinatorial Significance," Information and Control, vol. 23, no. 5, 1973, pp. 407--438. [10] J. M. Goethals, S. L. Snover, "Nearly Perfect Binary Codes," Discrete Math., 3, 1972, pp. 65-88. [11] J. M. Goethals, H. C. A. van Tilborg, "Uniformly packed codes," Philips Res. Repts 30, 1975, pp. 9-36. [12] M. Karpovsky, "Weight distribution of translates, covering radius and perfect codes correcting errors of the given multiplicities," IEEE Trans. Inform. Theory IT-27, 1981, pp. 462--472. (13] J. E. MacDonald, "Design methods for maximum minimum-distance error-correcting codes," IBM J. Res. Devel. vol. 4, 1960, pp. 43-57. [14] N. V. Semakov, V. A. Zinoviev, G. V. Zaitsev, "Uniformly Packed Codes," Problems of Inform. Transmission, vol. 7, 1971, no. 1, pp. 38-50. [15] Th. Skolem, P. Chowla, and D. J. Lewis, "The Diophantine equation 2n-2 -7 = x 2 and related problems," Proc. Amer. Math. Soc. 10 , 1959, pp. 663-669. [16] G. J. M. van Wee, G. D. Cohen, S. N. Litsyn, "A note on Perfect Multiple Coverings of Hamming Spaces," IEEE Trans. Inform. Theory, IT- 37, no. 3, 1991, pp. 678-682. 

On Perfect Weighted Coverings with Small Radius

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On Perfect Weighted Coverings with Small Radius

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