An Investigation of Linear Complementary Pairs of Codes over Finite Fields and Their Generalisations

Abstract

This comprehensive study delves into the realms of cyclic, constacyclic, quasi-cyclic, newlineand conjucyclic codes over finite fields. Going beyond a general examination, we immerse newlineourselves in the nuanced landscape of and#8467;-intersection pairs- a concept that extends newlinethe idea of linear complementary pairs of codes. Throughout this study, we meticulously newlineprovide various characterisations for and#8467;-intersection pairs, delving deep into their newlineproperties and shedding light on their nuanced analyses. To lend a comprehensive understanding newlineto our research, we showcase diverse applications stemming from the profound newlineimplications of and#8467;-intersection pairs within these code families. This not only newlineenriches the theoretical groundwork but also emphasizes the practical relevance and newlinepotential impact of our findings. newlineConjucyclic codes are underexplored codes within the broader family of codes that newlineinclude cyclic, constacyclic, and quasi-cyclic codes, etc. Despite their significance in newlinequantum error correction, conjucyclic codes have been relatively overlooked in the existing newlineliterature. Our focus centres on additive conjucyclic (ACC) codes over F4, a finite newlinefield of order 4. Employing a trace inner product, we derive the duals of ACC codes, newlinerevealing the trace hull and its dimension. Additionally, we establish a necessary and newlinesufficient condition for additive complementary dual (ACD) codes. Also, we identify a newlinenecessary condition for an additive complementary pair of conjucyclic codes over F4. newlineFurthermore, we construct entanglement-assisted quantum error-correcting (EAQEC) newlinecodes using the trace code of ACC codes. Shifting attention to conjucyclic codes over newlineFp2 , we observe the absence of non-trivial linear conjucyclic codes. This leads us to the newlinecharacterisation of and#8467;-intersection pair of additive conjucyclic codes. We observe that newlinethe largest Fp-subcode of an ACC code over Fp2 is cyclic. We determine its generating newlinepolynomial, enabling the calculation of the ACC code size. Additionally, we discuss newlinethe cyclic nature of the trace code of an A