Batch codes from Hamming and Reed-Muller codes

Year 2018, Volume: 5 Issue: 3, 153 - 165, 08.10.2018

Travis Baumbaugh Yariana Diaz Sophia Friesenhahn Felice Manganiello Alexander Vetter

https://doi.org/10.13069/jacodesmath.466634

Cited By: 1

Abstract

Batch codes, introduced by Ishai et al., encode a string $x \in \Sigma^{k}$ into an $m$-tuple of strings, called buckets. In this paper we consider multiset batch codes wherein a set of $t$-users wish to access one bit of information each from the original string. We introduce a concept of optimal batch codes. We first show that binary Hamming codes are optimal batch codes. The main body of this work provides batch properties of Reed-Muller codes. We look at locality and availability properties of first order Reed-Muller codes over any finite field. We then show that binary first order Reed-Muller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary Reed-Muller codes which have order less than half their length.

Keywords

Batch codes, Hamming codes, Reed-Muller codes

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