Intersection of linear and multi-twisted codes with applications
Ramy Takieldin, André Leroy
TL;DR
This work develops explicit generator-matrix methods for the intersection of linear codes and extends these ideas to the multi-twisted (MT) codes, using generator polynomial matrices (GPMs) as the primary algebraic tool. It provides a complete GPM-based framework for intersections, Galois dual intersections, and reversibility, including necessary and sufficient conditions for various MT-code properties such as self-orthogonality, dual containment, LCD, and reversibility. It proves that MT codes are closed under reversal and supplies explicit GPMs for reversed MT codes, as well as GPMs for intersections of MT codes with identical block lengths and shifts, and for MT codes with Galois duals. The results also identify conditions under which MT intersections retain MT structure, linking minimum distances, shift constants, and the lcm-based parameter N, and they yield practical criteria for containment and trivial intersections. Overall, the paper provides a comprehensive, GPM-centric toolkit for analyzing and constructing MT codes and their dualities with wide implications for cryptography and quantum error correction.
Abstract
In this paper, we derive a formula for constructing a generator matrix for the intersection of any pair of linear codes over a finite field. Consequently, we establish a condition under which a linear code has a trivial intersection with another linear code (or its Galois dual). Furthermore, we provide a condition for reversibility and propose a generator matrix formula for the largest reversible subcode of any linear code. We then focus on the comprehensive class of multi-twisted (MT) codes, which are naturally and more effectively represented using generator polynomial matrices (GPMs). We prove that the reversed code of an MT code remains MT and derive an explicit formula for its GPM. Additionally, we examine the intersection of a pair of MT codes, possibly with different shift constants, and demonstrate that this intersection is not necessarily MT. However, when the intersection admits an MT structure, we propose the corresponding shift constants. We also establish a GPM formula for the intersection of a pair of MT codes with the same shift constants. This result enables us to derive a GPM formula for the intersection of an MT code and the Galois dual of another MT code. Finally, we examine conditions for various properties on MT codes. Perhaps most importantly, the necessary and sufficient conditions for an MT code to be Galois self-orthogonal, Galois dual-containing, Galois linear complementary dual (LCD), or reversible.