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

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 C, identify for each dimension ν, the smallest size of the support of a subcode of 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 [1] 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 in [2]. This results in a characterization of the GHW hierarchy of binary PRM codes.

A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields

“…, − 1. On the other hand, the usual lexicographic order on Q corresponds to the total order on H ( ) , which we denote, as in (17), by ≺ lex . Note that if µ, ν ∈ H ( ) are such that ν | µ, i.e., ν divides µ, then ν lex µ.…”

“…Remark 7.7. In a recent work, Ramkumar, Vajha, and Vijay Kumar [17] have determined all the generalized Hamming weights of what they call the "binary projective Reed-Muller code". However, the code they consider is not PRM 2 (d, m) as defined above (and studied by Lachaud [15], Sørensen [19], and others), but, in fact, a puncturing of a subcode of PRM 2 (d, m).…”