Johnson bound

Abstract: Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following "curve-fitting" problem over a field F: given n points ((x/sub i//spl middot/y/sub i/))/sub i=1//sup n/, x/sub i/, y/sub i//spl isin/F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y/sub i/=p(x/sub i/) for all but at most e values of i/spl isin/(1,...,n). We give an algorithm that solves this problem for e 1/3, where the result yields the first asymptotic improvement in four decades. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometry codes. In both cases, we obtain a list decoding algorithm that corrects up to n-/spl radic/(n(n-d')) errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of Shokrollahi and Wasserman (see in Proc. 29th Annu. ACM Symp. Theory of Computing, p.241-48, 1998) and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve-fitting problem, which may be of use in soft-decision decoding algorithms for Reed-Solomon codes.

Abstract: A table of binary constant weight codes of length nl28 is presented. Explicit constructions are given for most of the 600 codes in the table; the majority of these codes are new. The known techniques for constructing constant weight codes are surveyed, and a table of (unrestricted) binary codes of length nl28 is given

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/spl les/24 and d/spl les/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/spl les/28 and d/spl les/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 constant-weight 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-bounded-weight 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.