Disproving some theorems in Sharma and Chauhan et al. (2018, 2021)
Ramy Takieldin, Patrick Solé
TL;DR
This work debunks key claimed LCD criteria for multi-twisted codes by presenting counterexamples to Sharma2018 and Chauhan2021 in both coprime and non-coprime settings, showing that code dimension alone cannot determine LCD, self-orthogonality, or dual containment. It then introduces a generalized, less restrictive sufficient condition (corr2) for an MT code to be LCD, under a weaker coprimality condition, and demonstrates that both the code and its dual decompose into direct sums of constacyclic codes under this framework. The authors further show that corr2 can identify more LCD MT codes than Sharma2018 (with examples illustrating its superiority) and provide a dimension formula for any $\rho$-generator MT code that does not require a normalized generating set. Overall, the paper advances the theory of MT codes by relaxing structural constraints, extending LCD criteria, and enabling direct computation of code dimensions via determinantal divisors.
Abstract
The main objective of this work is to show, through counterexamples, that some of the theorems presented in the papers of Sharma \textit{et al.} (2018) and Chauhan \textit{et al.} ( 2021) are incorrect. Although they used these theorems to establish a sufficient condition for a multi-twisted (MT) code to be linear complementary dual (LCD), we show that this condition itself remains valid. We further improve this condition by removing the restrictions on the shift constants and relaxing the required coprimality condition. We show that compared to the previous condition, the modified condition is able to identify more LCD MT codes. Furthermore, without the need for a normalized set of generators, we develop a formula to determine the dimension of any $ρ$-generator MT code.