Proceedings of the First French-Israeli Workshop on Algebraic Coding
IEEE Transactions on Information Theory
Minimal vectors in linear codes
IEEE Transactions on Information Theory
On cryptographic properties of the cosets of R(1, m)
IEEE Transactions on Information Theory
Random codes: minimum distances and error exponents
IEEE Transactions on Information Theory
Error-correction capability of binary linear codes
IEEE Transactions on Information Theory
The coset distribution of triple-error-correcting binary primitive BCH codes
IEEE Transactions on Information Theory
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The error correction capability of binary linear codes with minimum distance decoding, in particular the number of correctable/uncorrectable errors, is investigated for general linear codes and the first-order Reed-Muller codes. For linear codes, a lower bound on the number of uncorrectable errors is derived. The bound for uncorrectable errors with a weight of half the minimum distance asymptotically coincides with the corresponding upper bound for Reed-Muller codes and random linear codes. For the first-order Reed-Muller codes, the number of correctable/uncorrectable errors with a weight of half the minimum distance plus one is determined. This result is equivalent to deriving the number of Boolean functions of m variables with nonlinearity 2m-2 + 1 The monotone error structure and its related notions larger half and trial set, which were introduced by Helleseth, Kløve, and Levenshtein, are mainly used to derive the results.