Quasi-twisted codes as contractions of quasi-cyclic codes (Q6548440)

Let \(\ell\) be a positive integer. A linear code \(C\subseteq \mathbb{F}_q^{m\ell}\) is called a \(\lambda\)-quasi-twisted (\(\lambda\) -QT) code of index \(\ell\) and co-index \(m\) if it is invariant under the \(\lambda\)-constashift of codewords by \(\ell\) positions. In particular, if \(\ell = 1,\) then \(C\) is a \(\lambda\)-constacyclic code, and if \(\lambda= 1\) or \(q = 2,\) then \(C\) is a quasi-cyclic (QC) code of index \(\ell.\)\N\NIn this study, the authors aim at generalizing Bierbrauer's contraction results to quasi-cyclic and quasi-twisted codes. Using the concatenated structure and the trace representation of codewords, it is shown that any given quasi-twisted code can be extended to a quasi-cyclic code with the same constituents, where the dimensions of both codes are equal and the extension in the length and minimum distance is determined by the multiplicative order of the shift constant \(\lambda.\) It is shown also, conversely, that quasi-cyclic codes with a certain concatenated structure can be contracted to a quasi-twisted code of the same dimension, where the length and the minimum distance is divided by the same amount. Some examples of the best known quasi-cyclic and quasi-twisted codes that satisfy the requirements of the main result, are given. When \(m = 4\), \(\ell=7,\) \(q = 3,\) a particular ternary 2-QT code results in a \([28, 4, 18]_3\) code, attaining the best-known minimum distance with length 28 and dimension 4. The best-known minimum distance codes with dimension 4 and lengths 80 and 160, as well as the best-known minimum distance codes with dimension 3 and lengths 42 and 126, are also included as an illustration of the usefulness of this construction.