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

TL;DR

This paper characterizes the entire hierarchy of Generalized Hamming Weights for binary Projective Reed-Muller codes, providing exact bounds crucial for assessing code security and complexity.

Contribution

It derives a matching lower bound for the GHW of binary PRM codes, completing the characterization of their GHW hierarchy.

Findings

Exact GHW hierarchy for binary PRM codes established

Matching lower bounds derived using adapted Wei's techniques

Implications for code security and trellis complexity analyzed

Abstract

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 $\nu$, the smallest size of the support of a subcode of ${\cal C}$ of dimension $\nu$. 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.