Redundancies of Correction-Capability-Optimized Reed-Muller Codes

This article is focused on some variations of Reed-Muller codes that yield improvements to the rate for a prescribed decoding performance under the Berlekamp-Massey-Sakata algorithm with majority voting. Explicit formulas for the redundancies of the new codes are given. Introduction Reed-Muller codes belong to the family of evaluation codes, commonly defined on an order domain. The decoding algorithm widely used for evaluation codes is an adaptation of the Berlekamp-Massey-Sakata algorithm together with the majority voting algorithm of Feng-Rao-Duursma. By analyzing majority voting, one realizes that only some of the parity checks are really necessary to perform correction of a given number of errors. New codes can be defined with just these few checks, yielding larger dimensions while keeping the same correction capability as standard codes [4, 5]. These codes are often called Feng-Rao improved codes. A different improvement to standard evaluation codes is given in [8]. The idea is that under the Berlekamp-Massey-Sakata algorithm with majority voting, error vectors whose weight is larger than half the minimum distance of the code are often correctable. In particular this occurs for generic errors (also called independent errors in [9, 6]), whose technical algebraic definition can be found in the mentioned references. Generic errors of weight t can be a very large proportion of all possible errors of weight t, as in the case of the examples worked out in [8]. This suggests that a code be designed to correct only generic errors of weight t rather than all error words of weight t. Using this restriction, one obtains new codes with much larger dimension than that of standard evaluation codes correcting the same number of errors. Universitat Autònoma de Barcelona, [email protected] San Diego State University, [email protected]