Vinayak Ramkumar — Page 2

arXiv2018

Determining the Generalized Hamming Weight Hierarchy of the Binary Projective Reed-Muller Code

Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code ${\cal C}$, identify for each dimension $ν$, the smallest size of the support of a subcode of ${\cal C}$ of dimension $ν$. The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work by the same authors, a code-shortening algorithm was employed to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM) codes. In the present paper, we derive a matching lower bound by adapting the proof techniques used originally for Reed-Muller (RM) codes by Wei. This results in a characterization of the GHW hierarchy of binary PRM codes.

cs.IT↗

arXiv2017

Binary, Shortened Projective Reed Muller Codes for Coded Private Information Retrieval

The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with replicated-server PIR. In the present paper, the construction of an $(n,k)$ $τ$-server binary, linear PIR code having parameters $n = \sum\limits_{i = 0}^{\ell} {m \choose i}$, $k = {m \choose \ell}$ and $τ= 2^{\ell}$ is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary, Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any $τ$ of the form $2^{\ell}$, $2^{\ell}-1$ and any value of $k$, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with other PIR codes appearing in the literature. For the particular case of $τ=3,4$, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length $n(k, τ)$ of a systematic PIR code. It follows from a result by Vardy and Yaakobi, that these codes also yield optimal, systematic primitive multi-set $(n, k, τ)_B$ batch codes for $τ=3,4$. The PIR code constructions presented here also yield upper bounds on the generalized Hamming weights of binary PRM codes.

cs.IT↗