1408 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 4, APRIL 2005 Error-Correction Capability of Binary Linear Codes Tor Helleseth, Fellow, IEEE, Torleiv Kløve, Fellow, IEEE, and Vladimir I. Levenshtein, Fellow, IEEE Abstract—The monotone structure of correctable and uncor- rectable errors given by the complete decoding for a binary linear code is investigated. New bounds on the error-correction capability of linear codes beyond half the minimum distance are presented, both for the best codes and for arbitrary codes under some restrictions on their parameters. It is proved that some known codes of low rate are as good as the best codes in an asymptotic sense. Index Terms—Error-correction capability, linear codes, min- imal words, monotone functions, test set, trial set, Reed–Muller codes. I. INTRODUCTION I N complete (maximum-likelihood) decoding of linear codes, there is some freedom in the choice of which errors will be corrected. More precisely, the correctable errors are exactly the coset leaders, and when there is more than one vector of min- imum weight in a coset, any one of them can be selected as the coset leader. It has long been known that if the lexicographi- cally smallest minimum-weight vectors are chosen as the coset leaders, then the set of correctable errors (coset leaders) gets a monotone structure: if is a correctable error and (that is, for all ), then is also a correctable error. This has been regarded as a nice property, but without much further relevance. However, Zemor [25] has studied some important consequences of this property. The goal of this paper is to argue that it is, in fact, a fundamental property with a number of important impli- cations, and we study some of these implications. The paper is organized as follows. Section II is a short introduction to the complete decoding and describes the monotone structure of the sets of correctable and uncorrectable errors and also introduces some notations. In particular, we introduce trial sets of codewords, considering the linear ordering of all vectors of a fixed length defined by if and only if has smaller Hamming weight than or they have the same Hamming weight but is lexicographically smaller than . In Section III, we describe the minimal uncorrectable errors under the ordering . For any such vector, we determine all vec- tors in its coset which precede it in the ordering . This allows us to prove that all minimal uncorrectable errors are so-called larger halves of minimal codewords. Further, we use the de- scription to give a gradient-like decoding algorithm based on Manuscript received December 2, 2002; revised October 4, 2004. This work was supported by the Norwegian Research Council; the work of V. Levenshtein was also supported by the Russian Foundation for Basic Research under Grant 02-01-00687. T. Helleseth and T. Kløve are with the Department of Informatics, University of Bergen, N-5020 Bergen, Norway. V. I. Levenshtein is with the Keldysh Institute for Applied Mathematics, RAS, 125047 Moscow, Russia. Communicated by R. Koetter, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2005.844080 trial sets of codewords. Finally, we give improved estimates of the number uncorrectable errors of any weight larger than half the minimum distance. In Section IV, we consider , the fraction of all errors of weight that are correctable. A consequence of the mono- tonicity of the set of correctable errors and a well-known result on monotone sets is that, for any in the range from half the minimum distance to the covering radius, decreases with the growing . Therefore, we also have a well-defined “inverse” , the error-correction capability function, that is defined as the largest such that . The rest of the paper is devoted to the study of these two functions, in particular, the asymptotic values for infinite sequences of codes. In our investigation of the quantity plays a significant role. Note that is a necessary condition for the existence of a -error-correcting code (the Hamming bound). The monotonicity of implies that for any code and any and hence as . On the other hand, using random selection on a set of codes we prove, for any , , and , the existence of an code for which Therefore, for such a code, as These results allows us to obtain in Section V precise esti- mates of for the best codes with for a fixed , . We give two explicit positive constants and such that for any there exists an code with such that and that for any code with and any fixed , if and is sufficiency large. (Here and later, , , is the parameter which is uniquely defined by the equation where is the Shannon entropy.) We also study sequences of codes where the rate goes to zero. For sequences of codes where and for which for a fixed and , 0018-9448/$20.00 © 2005 IEEE