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<p>The main objective of this work is to show, through counterexamples, that some of the theorems presented in the papers of Sharma et al. (2018) and Chauhan 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\rho\)</EquationSource> </InlineEquation>-generator MT code. On February 21, 2025, we contacted Sharma et al. to inform them of these inaccuracies and provided them with a copy of our version uploaded to http://arXiv:2503.03762.</p>

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Disproving and strengthening some theorems in Sharma and Chauhan et al. (2018, 2021)

  • 期刊论文

  • 发表日期：

    2025年11月17日

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  • Ramy Taki Eldin,
  • Patrick Solé

摘要

The main objective of this work is to show, through counterexamples, that some of the theorems presented in the papers of Sharma et al. (2018) and Chauhan 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 \(\rho\) -generator MT code. On February 21, 2025, we contacted Sharma et al. to inform them of these inaccuracies and provided them with a copy of our version uploaded to http://arXiv:2503.03762.