Minimum distance functions of graded ideals and Reed-Muller-type codes

The List-Decoding Size of Reed-Muller Codes

In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman [4] on the list size of Reed-Muller codes apply only up to the minimum distance of the code. In this w...

Correctable Errors of Weight Half the Minimum Distance for the First-Order Reed-Muller Codes

The number of correctable errors of weight half the minimum distance is given for the first-order binary Reed-Muller codes. It is shown that the size of trial set, which is introduced by Helleseth et al. and can be used for a minimum distance decoding or estimating the number of uncorrectable errors, for the first-order Reed-Muller codes is at least that of minimal codewords except for small code

Correctable Errors of Weight Half the Minimum Distance Plus One for the First-Order Reed-Muller Codes

The number of correctable/uncorrectable errors of weight half the minimum distance plus one for the first-order Reed-Muller codes is determined. From a cryptographic viewpoint, this result immediately leads to the exact number of Boolean functions of m variables with nonlinearity 2m−2 + 1. The notion of larger half and trial set , which is introduced by Helleseth, Kløve, and Levenshtein to desc...

The shift bound for cyclic, Reed-Muller and geometric Goppa codes

We give a generalization of the shift bound on the minimum distance for cyclic codes which applies to Reed-Muller and algebraicgeometric codes. The number of errors one can correct by majority coset decoding is up to half the shift bound. 1991 Mathematics Subject Classification: 94B27, 14H45.

Decoding Reed-Muller Codes beyond Half the Minimum Distance