Ranks and Kernels of Codes from Generalized Hadamard Matrices

TL;DR

This paper investigates the properties of generalized Hadamard matrices, establishing their connection to self-orthogonal codes, and provides bounds and constructions for their ranks and kernels.

Contribution

It proves that certain generalized Hadamard matrices generate self-orthogonal codes and offers bounds and explicit constructions for their ranks and kernels.

Findings

Generalized Hadamard matrices over certain fields generate self-orthogonal codes.

Bounds are established for the rank and kernel dimension of these codes.

Explicit constructions are provided for specific rank and kernel dimensions.

Abstract

The ranks and kernels of generalized Hadamard matrices are studied. It is proven that any generalized Hadamard matrix $H(q,\lambda)$ over $F_q$, $q>3$, or $q=3$ and $\gcd(3,\lambda)\not =1$, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes. For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed.