On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

TL;DR

This paper explores the relationships between Pless symmetry codes, ternary QR codes, Hadamard matrices, and related combinatorial designs, providing classification results and automorphism group analyses for specific cases.

Contribution

It establishes new connections between certain symmetry codes and Hadamard matrices, classifies all Hadamard matrices derived from a specific code, and analyzes automorphism groups of associated designs.

Findings

Full automorphism groups of certain codes and designs are identified.

All Hadamard matrices of order 36 from a specific code are classified into two classes.

Automorphism groups vary significantly between different Hadamard matrices and associated designs.

Abstract

It is proved that a code $L(q)$ which is monomially equivalent to the Pless symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ associated with a Paley-Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley-Hadamard matrix of type I. If $q=5, 11, 17, 23$, then the full permutation automorphism group of $L(q)$ coincides with the full automorphism group of $D(q)$, and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code $C(17)$ are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley-Hadamard matrix $H$ of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix $H'$ such that the symmetric 2-$(36,15,6)$ design $D$ associated with $H'$ has trivial full automorphism group, and the incidence matrix of $D$ spans a ternary code equivalent to $C(17)$.