Batch Codes from Hamming and Reed-Müller Codes

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-M\"uller codes. We look at locality and availability properties of first order Reed-M\"uller codes over any finite field. We then show that binary first order Reed-M\"uller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary Reed-M\"uller codes which have order less than half their length.