On the classification of self-dual [20,10,9] codes over GF(7)

TL;DR

This paper classifies a unique self-dual [20,10,9] code over GF(7) by linking it to unimodular lattices and skew-Hadamard matrices, providing a complete characterization of such codes.

Contribution

It establishes the uniqueness of a specific self-dual code over GF(7) associated with the lattice D_{20}^+ and connects code classification to skew-Hadamard matrices.

Findings

The extended quadratic residue code is unique up to equivalence.

The code's lattice is isomorphic to D_{20}^+.

Classification reduces to skew-Hadamard matrices of order 20.

Abstract

It is shown that the extended quadratic residue code of length 20 over GF(7) is a unique self-dual [20,10,9] code C such that the lattice obtained from C by Construction A is isomorphic to the 20-dimensional unimodular lattice D_{20}^+, up to equivalence. This is done by converting the classification of such self-dual codes to that of skew-Hadamard matrices of order 20.