Odd prime codes differ from higher power linear codes in structure
Nonequivalence of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear and $\mathbb{Z}_{p^s}$-linear generalized Hadamard codes for odd primes
Information Theory
Summary
This work compares two advanced types of error-correcting codes that help detect and fix mistakes in data. The authors find that for odd prime numbers, certain codes built from smaller building blocks behave differently than codes built over larger number systems, meaning they are fundamentally different. They also show when codes built this way match or don't match each other and count how many unique codes exist. This helps understand the diversity and properties of these complex code families.
What this means in practice
- •For communication engineers: Choose appropriate code families for error correction in digital communication when working with codes over prime power alphabets.
- •For coding theory developers: Avoid assuming equivalence between certain code constructions when designing new coding schemes over modular integer rings for robust data transmission.
A theory result. No direct application yet.
Authors
Dipak K. Bhunia
Abstract
The $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive codes are subgroups of $\mathbb{Z}_p^{α_1}\times\mathbb{Z}_{p^2}^{α_2}$, and they can be seen as linear codes over $\mathbb{Z}_p$ when $α_2=0$, as $\mathbb{Z}_{p^2}$-additive codes when $α_1=0$, and as $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $p=2$. A $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code. For every prime $p$, there are recursive constructions of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear GH codes $H_p^{t_1,t_2}$ with $α_1\neq0$ and $α_2\neq0$, and of $\mathbb{Z}_{p^s}$-linear GH codes $\bar H_p^{t_1,\dots,t_s}$. We show that the kernel blocks of a nonlinear code $H_p^{t_1,t_2}$ give equivalent punctured codes, and that these punctured codes are linear. Any of them is called the derived code of $H_p^{t_1,t_2}$. We also determine the ranks of the codes $H_p^{t_1,t_2}$ and $\bar H_p^{t_1,t_2}$. For odd $p$, the rank of $H_p^{t_1,t_2}$ depends on the Bernoulli numbers modulo $p$. By comparing the derived codes and the ranks, we show that, for odd $p$, no code $H_p^{t_1,t_2}$ is equivalent to any of these $\mathbb{Z}_{p^s}$-linear GH codes. For $p=2$, a nonlinear code $H_2^{t_1,t_2}$ is equivalent to a $\mathbb{Z}_{2^s}$-linear Hadamard code if and only if $t_2\geq2$. Finally, we count the nonequivalent codes in the union of both families.