Upper bounds for fixed-weight codes of specified minimum distance (Corresp.)

Freiman, C.

doi:10.1109/tit.1964.1053677

Cited by 13 publications

(4 citation statements)

References 2 publications

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“…The inequalities (29) and (30) in Corollary 6 appear to be new, whereas (28) was found previously by both Levenshtein 39] and Johnson 37]. They use this inequality for all b > 0 (see also Section V-A).…”

Section: B New Boundsmentioning

confidence: 72%

Upper bounds for constant-weight codes

Agrell

Vardy

Zeger

2000

IEEE Trans. Inform. Theory

139

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| Let A(n; d; w) denote the maximum possible number of codewords in an (n; d; w) constant-weight binary code. We improve upon the best known upper bounds on A(n; d; w) in numerous instances for n 6 24 and d 6 12, which is the parameter range of existing tables. Most improvements occur for d = 8; 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n 6 28 and d 6 14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n; d; w) by means of mapping constantweight codes into Euclidean space. This approach produces, among other results, a bound on A(n; d; w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doubly-constant-weight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doubly-boundedweight codes, which may be thought of as a generalization of the doubly-constant-weight codes. Subsequently, a class of Euclidean-space codes, called zonal codes, is introduced, and a bound on the size of such codes is established. This is used to derive bounds for doubly-bounded-weight codes, which are in turn used to derive bounds on A(n; d; w). We also develop a universal method to establish constraints that augment the Delsarte inequalities for constant-weight codes, used in the linear programming bound. In addition, we present a detailed survey of known upper bounds for constant-weight codes, and sharpen these bounds in several cases. All these bounds, along with all known dependencies among them, are then combined in a coherent framework that is amenable to analysis by computer. This improves the bounds on A(n; d; w) even further for a large number of instances of n, d, and w. Keywords| Constant-weight codes, Delsarte inequalities, doubly-bounded-weight codes, doubly-constant-weight codes, linear programming, spherical codes, zonal codes. I. Introduction A N (n; d; w) constant-weight binary code is a set of binary vectors of length n, such that each vector contains w ones and n w zeros, and any two vectors di er in at least d positions. Given the three parameters: length n, weight w, and distance d, what is the largest possible size A(n; d; w) of an (n; d; w) constant-weight binary code? This question has been studied for almost four decades, and remains one of the most basic questions in coding theory. Although the general answer is not known, various upper and lower bounds on A(n; d; w) have been developed.

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Section: B New Boundsmentioning

confidence: 72%

Upper bounds for constant-weight codes

Agrell

Vardy

Zeger

2000

IEEE Trans. Inform. Theory

139

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“…In the following theorem, we picked feasible points by judiciously choosing constrained spaces that contain S and estimating the corresponding ball sizes. This is motivated by Freiman's and Berger's methods [10], [11] that improve the usual sphere-packing bounds for constant weight codes.…”

Section: Generalized Sphere-packing Boundmentioning

confidence: 99%

Generalized Sphere-Packing Bound for Subblock-Constrained Codes

Kiah

Tandon

Motani

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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We apply the generalized sphere-packing bound to two classes of subblock-constrained codes.À la Fazeli et al. (2015), we made use of automorphism to significantly reduce the number of variables in the associated linear programming problem. In particular, we study binary constant subblock-composition codes (CSCCs), characterized by the property that the number of ones in each subblock is constant, and binary subblock energy-constrained codes (SECCs), characterized by the property that the number of ones in each subblock exceeds a certain threshold. For CSCCs, we show that the optimization problem is equivalent to finding the minimum of N variables, where N is independent of the number of subblocks. We then provide closed-form solutions for the generalized spherepacking bounds for single-and double-error correcting CSCCs. For SECCs, we provide closed-form solutions for the generalized sphere-packing bounds for single errors in certain special cases. We also obtain improved bounds on the optimal asymptotic rate for CSCCs and SECCs, and provide numerical examples to highlight the improvement.Example 4 (Example 3 continued). Consider m = 4, L = 3, w = 2 and t = 1. Then consider the pair Y * , X * , where Y * = (0, 1/12, 1/4, 1/4, 0, 0, 0, 0, 0), and X * = (1, 0, 0, 18, 45/2).

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“…Upper bounds for T(p, q, r) (which is important in studying errorcorrecting codes) have been given by Johnson [1962], [1963], [1971], [1972], Freiman [1964], Sch~inheim [196@ Berger [1967], Niven [1970], and Levengtein [1971]. Clearly, Theorem 3 may be strengthened to the assertion t (p) <~ t (p) <~ r (p).…”

Section: T(p)=[[(p-1)/2]p/3]-imentioning

confidence: 99%