A generalization of quasi-twisted codes: Multi-twisted codes

Abstract

Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this work we introduce a new generalization of QT codes that we call multi-twisted (MT) codes and study some of their basic properties. Presenting several methods of constructing codes in this class and obtaining bounds on the minimum distances, we show that there exist codes with good parameters in this class that cannot be obtained as QT or constacyclic codes. This suggests that considering this larger class in computer searches is promising for constructing codes with better parameters than currently best-known linear codes. Working with this new class of codes motivated us to consider a problem about binomials over finite fields and to discover a result that is interesting in its own right.

Good Classical and Quantum Codes from Multi-Twisted Codes

“…Our work on this also led to a couple of theoretical results about the binomials of the form x n − a over F q . The first one (Theorem 5.3 below) is similar to Theorem 4.1 in [4]. We start with the results on binomials.…”

“…Hundreds of BKLCs have been obtained, usually by computer searches, from the class of QT codes. More recently, a generalization of QT codes called multi-twisted (MT) codes was introduced in [4]. Therefore, MT codes are a generalization of cyclic, negacyclic, consta-cyclic (CC), quasi-cyclic (QC), generalized quasi-cyclic (GQC) and quasi-twisted codes (QT).…”

Polycyclic Codes Associated with Trinomials: Good Codes and Open Questions

“…Note that gcdpx n ´ax i ´b, x n ´cx i ´dq " gcdpx n ´ax i ´b, x n ´ax i ´b ṕx n ´cx i ´dqq " gcdpx n ´ax i ´b, pc ´aqx i `pd ´bqq. Using a similar argument, we have the following, gcdpx n ´ax i ´b, x n ´cx i ´dq " gcdpx n ´ax i ´b, pc ´aqx i `pd ´bqq " gcdpx n ´ax i ´b, x i `pd ´bqpc ´aq ´1q " gcdpx n ´ax i ´b `a ¨px i `pd ´bqpc ´aq ´1q, x i `pd ´bqpc ´aq ´1q " gcdpx n `apd ´bqpc ´aq ´1 ´b, x i `pd ´bqpc ´aq ´1q Therefore, by Theorem 4.1 in [4] gcdpx n ´ax i ´b, x n ´cx i ´dq is either 1 or a binomial of degree gcdpn, iq. Lemma 8.4.…”

Skew-Constacyclic Codes Over

“…We generalize the notion of multi-twisted codes which be defined in [3], by introducing skew-multi-twisted codes. Definition 2.…”