Upper bounds for constant-weight codes

Abstract

| 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.

Defect-tolerant demultiplexer circuits based on threshold logic and coding

“…In our scheme, 'stuck open' defects will cause a given cross-wire to fail with probability approximately [24]. On the other hand, defects of type 'closed' will cause a given cross-wire in [24] to fail with probability w p 2 c ≈ (2.8 p c ) 2 , compared to 1 2 w(w−1) p 2 c ≈ (7.4 p c ) 2 in our scheme if only one defect of type S-closed is tolerated per cross-wire, or to 1 6 w(w−1)(w−2) p 3 c ≈ (5.5 p c ) 3 if two such defects are tolerated.…”

“…Reference [6] contains constructions of constant-weight codes and bounds on their sizes for a wide range of values n, w, and d (see also [1]). As can be seen in the tables in [6], the largest constant-weight code for given values of n and d is attained therein for w ∈ { n/2 , n/2 } (but we are unaware of a proof that establishes this phenomenon generally).…”

The number of olfactory stimuli that humans can discriminate is still unknown

“…This is well-known from the theory of error-correcting codes, where Equations 5, 6 represent lower and upper bounds on the solution to a homologous problem in coding theory (MacWilliams and Sloane, 1977). These bounds are essentially the constantweight versions of the Gilbert-Varshamov and Hamming bounds, respectively, and have been proven mathematically; specifically, the lower bound is due to Levenshtein (Levenshtein, 1971;Jiang and Vardy, 2004), and the upper bound is slightly weaker version of that developed by Freiman, Berger, and Johnson (Freiman, 1964;Agrell et al, 2000).…”

Estimating the indivisible error detecting сodes based on an average probability method

“…This code consists of k 1's and n-k 0's where n is the length of the code combination. Identifying the error is the discrepancy between the number of n-k 0's and the number of 1's k in the code combination [12]. Study [13] assessed the probability of an undetectable error in binary (n, 26, m) non-linear codes with a permanent weight using a binary symmetrical channel.…”