Construction and Classification of Generalized Hadamard Codes over Gaussian Integer Rings $\mathbb{Z}_{3^s}[i]$

Abstract

This paper provides a general guideline on how to construct Generalized Hadamard (GH) codes over Gaussian integer rings $\mathbb{Z}_{3^s}[i]$, where $i^2 + 1 = 0$. We define an algebraic basis with the help of the Gaussian integers and construct a special Gray map which converts the codes of these rings into ternary image representations over $\mathbb{F}_3[i] \cong \mathbb{F}_9$. By making a careful analysis, we derive conditions (needed and must-have) to ascertain the linearity properties of the GH codes linear over $\mathbb{Z}_{3^s}[i]$. The structural properties of these codes are discussed in more detail, such as their kernel architecture, rank specifications and combinatorial invariants. Moreover, we give a systematic code family of $\mathbb{Z}_{3^s}[i]$-linear Hadamard codes in terms of their algebraic and combinatorial characteristics. Our results generalize the classical GH code theory to new fields of algebra and have possible uses in secure communications and high-reliability systems of data transmission.