Hadamard full propelinear codes with associated group $C_{2t}\times C_2$; rank and kernel

TL;DR

This paper introduces a class of Hadamard full propelinear codes with a specific group structure, analyzing their rank and kernel, and establishing links to circulant Hadamard matrices, including a new example of order 16.

Contribution

It defines Hadamard full propelinear codes with group $C_{2t} imes C_2$, studies their properties, and links them to circulant Hadamard matrices, providing new examples and bounds.

Findings

Kernel dimension bounded by 3 for nonlinear codes

Established equivalence with circulant complex Hadamard matrices

Constructed a new circulant complex Hadamard matrix of order 16

Abstract

We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is $C_{2t}\times C_2$. We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by 3 if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order 16.