The two-sided Galois duals of multi-twisted codes

TL;DR

This paper characterizes the Galois duals of multi-twisted codes, introduces the concept of two-sided Galois duals, and provides formulas and conditions for their structure and relationships.

Contribution

It is the first to analyze both right and left Galois duals of multi-twisted codes simultaneously, defining the two-sided Galois dual and characterizing when it is also multi-twisted.

Findings

The Euclidean dual of a multi-twisted code is also multi-twisted.

Formulas for the generator polynomial matrix of the Galois dual are derived.

Conditions for the intersection and equality of right and left Galois duals are established.

Abstract

Characterizing the duals of linear codes with rich algebraic structures received great interest in recent decades. The beginning was by representing cyclic codes over finite fields as ideals in the polynomial ring. Subsequently, studying the duals of constacyclic, quasi-cyclic, quasi-twisted, generalized quasi-cyclic, and multi-twisted codes appeared extensively in literature. We consider the class of multi-twisted (MT) codes because it extends to all of these codes. We describe a MT code $\mathcal{C}$ as a module over a principal ideal domain. Hence, $\mathcal{C}$ has a generator polynomial matrix (GPM) that satisfies an identical equation. The reduced GPM of $\mathcal{C}$ is the Hermite normal form of its GPM. We show that the Euclidean dual $\mathcal{C}^\perp$ of $\mathcal{C}$ is MT as well. We prove a formula for a GPM of $\mathcal{C}^\perp$ using the identical equation of the reduced GPM of $\mathcal{C}$. Then we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Our study is the first to contain the right and left Galois duals of a linear code simultaneously. This gives two advantages: establishing their interconnected identities and introducing the two-sided Galois dual that has not previously appeared in the literature. We use a condition for the two-sided Galois dual of a MT code to be MT, hence its GPM is characterized. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. The latter case is considered for any linear code, where a necessary and sufficient condition is established for the equality of the right and left Galois duals.