Linear complementary pairs of skew constacyclic codes
F. J. Lobillo, José Manuel Muñoz
TL;DR
This work develops a framework for linear complementary pairs (LCPs) of skew $\lambda$-constacyclic codes, linking duality, factorization, and algebraic structure to design codes with prescribed security parameters. It extends BCH-like ideas to the skew constacyclic setting, providing explicit constructions for $\mathcal{C}=\mathcal{R}g$ and $\mathcal{D}=\mathcal{R}h$ that form LCPs with designed distances, and leverages duality via the anti-isomorphism $\Theta$ to analyze and control the dual distance. A key contribution is the use of norm maps, Wedderburn polynomial decompositions, and weight-preserving isomorphisms to generate numerous LCPs with fixed security parameters from a single construction, while preserving the minimum distance properties. The results yield a practical, algebraically rich toolkit for secure coding in hardware, enabling systematic construction of LCPs with high dual distance and flexible parameters. Overall, the paper combines skew polynomial techniques, BCH-like design principles, and automorphism-induced isometries to advance secure LCP design in the noncommutative code framework.
Abstract
Linear complementary pairs (LCPs) of codes have been studied since they were introduced in the context of discussing mitigation measures against possible hardware attacks to integrated circuits. In this situation, the security parameters for LCPs of codes are defined as the (Hamming) distance and the dual distance of the codes in the pair. We study the properties of LCPs of skew constacyclic codes, since their algebraic structure provides tools for studying their duals and their distances. As a result, we give a characterization for those pairs, as well as multiple results that lead to constructing pairs with designed security parameters. We extend skew BCH codes to a constacyclic context and show that an LCP of codes can be immediately constructed from a skew BCH constacyclic code. Additionally, we describe a Hamming weight-preserving automorphism group in the set of skew constacyclic codes, which can be used for constructing LCPs of codes.