Classification of generalized Hadamard matrices H(6,3) and quaternary Hermitian self-dual codes of length 18

TL;DR

This paper classifies all generalized Hadamard matrices H(6,3) of order 18 and quaternary Hermitian self-dual codes of length 18, providing a comprehensive enumeration and establishing their equivalence classes.

Contribution

It introduces a complete classification of H(6,3) matrices and Hermitian self-dual codes of length 18, linking combinatorial designs with coding theory.

Findings

85 generalized Hadamard matrices H(6,3) up to monomial equivalence

245 inequivalent Hermitian self-dual codes of length 18 over GF(4)

Two enumeration methods linking designs and codes

Abstract

All generalized Hadamard matrices of order 18 over a group of order 3, H(6,3), are enumerated in two different ways: once, as class regular symmetric (6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of order 3 acting semi-regularly on points and blocks, and secondly, as collections of full weight vectors in quaternary Hermitian self-dual codes of length 18. The second enumeration is based on the classification of Hermitian self-dual [18,9] codes over GF(4), completed in this paper. It is shown that up to monomial equivalence, there are 85 generalized Hadamard matrices H(6,3), and 245 inequivalent Hermitian self-dual codes of length 18 over GF(4).