Two-sided Galois duals of multi-twisted codes

Abstract

Galois duals of Multi-twisted (MT) codes are considered in this study. We describe a MT code mathcal {C} as a module over a principal ideal domain. Hence, mathcal {C} has a generator polynomial matrix (GPM) textbf{G} that satisfies an identical equation. We prove a GPM formula for the Euclidean dual mathcal {C}^perp using the identical equation of the Hermite normal form of textbf{G}. Next, 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. Some interconnected identities for the right and left Galois duals of a linear code are established and we also introduce the two-sided Galois dual. We use a condition that makes the two-sided Galois dual of a MT code MT, then we describe its GPM. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. A necessary and sufficient condition is established for the equality of the right and left Galois duals of any linear code.