New Set of Codes for the Maximum-Likelihood Decoding Problem

The maximum-likelihood decoding problem is known to be NP-hard for general linear and Reed-Solomon codes. In this paper, we introduce the notion of A-covered codes, that is, codes that can be decoded through a polynomial time algorithm A whose decoding bound is beyond the covering radius. For these codes, we show that the maximum-likelihood decoding problem is reachable in polynomial time in the code parameters. Focusing on bi- nary BCH codes, we were able to find several examples of A-covered codes, including two codes for which the maximum-likelihood decoding problem can be solved in quasi-quadratic time.Comment: in Yet Another Conference on Cryptography, Porquerolle : France (2010

Covering bb-Symbol Metric Codes and the Generalized Singleton Bound

Symbol-pair codes were proposed for the application in high density storage systems, where it is not possible to read individual symbols. Yaakobi, Bruck and Siegel proved that the minimum pair-distance of binary linear cyclic codes satisfies $d_2 \geq \lceil 3d_H/2 \rceil$ and introduced $b$-symbol metric codes in 2016. In this paper covering codes in $b$-symbol metrics are considered. Some examples are given to show that the Delsarte bound and the Norse bound for covering codes in the Hamming metric are not true for covering codes in the pair metric. We give the redundancy bound on covering radius of linear codes in the $b$-symbol metric and give some optimal codes attaining this bound. Then we prove that there is no perfect linear symbol-pair code with the minimum pair distance $7$ and there is no perfect $b$-symbol metric code if $b\geq \frac{n+1}{2}$. Moreover a lot of cyclic and algebraic-geometric codes are proved non-perfect in the $b$-symbol metric. The covering radius of the Reed-Solomon code in the $b$-symbol metric is determined. As an application the generalized Singleton bound on the sizes of list-decodable $b$-symbol metric codes is also presented. Then an upper bound on lengths of general MDS symbol-pair codes is proved.Comment: 21 page

On the decoder error probability for Reed-Solomon codes

Upper bounds On the decoder error probability for Reed-Solomon codes are derived. By definition, "decoder error" occurs when the decoder finds a codeword other than the transitted codeword; this is in contrast to "decoder failure," which occurs when the decoder fails to find any codeword at all. These results imply, for example, that for a t error-correcting Reed-Solomon code of length q - 1 over GF(q), if more than t errors occur, the probability of decoder error is less than 1/t!

On deep holes of generalized Reed-Solomon codes

Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word $u$ is a deep hole of the standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. In this paper, we extend this result by giving a new class of deep holes of the generalized Reed-Solomon codes.Comment: 5 page