Weight enumeration of codes from finite spaces

TL;DR

This paper investigates the weight enumerators of q-ary Simplex and Reed-Muller codes using geometric methods, providing explicit calculations and characterizations of subcode supports within finite projective and affine spaces.

Contribution

It introduces a geometric approach to compute weight enumerators and characterizes subcode supports for these classical codes, extending previous algebraic methods.

Findings

Explicit formulas for generalized and extended weight enumerators.

Complete characterization of subcode support sets.

Connection between geometric structures and code properties.

Abstract

We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a finite projective or affine space. As a result from the geometric method we use for the weight enumeration, we also completely determine the set of supports of subcodes and words in an extension code.