New quantum codes from constacyclic codes over the ring \(R_{k,m}\)
For positive integers \(k,m\) we define the ring \(R_{k,m}=\mathbb{F}_{p^m}[u_1,\ldots,u_k]/\langle u_i^2-1,u_iu_j-u_ju_i\rangle\) and it is a commutative ring of characteristics \(p\) and order \(p^{2^km}.\) For given element \(\gamma\in R_{k,m}^\ast\) a linear code \(C\) is \(\gamma\)-constacyclic if \(\tau_\gamma(c)=(\gamma c_{n-1},c_0,\ldots,c_{n-2})\in C\) for \(c=(c_0,\ldots,c_{n-1})\in C.\) The authors study the constacyclic codes over new and different non-chain rings in an attempt to construct more new quantum codes. In contrast to others, the authors have used here the Gray map \(\psi\) as the multiplication of a vector by an invertible matrix of order \(2^k\) which has the advantage to enhance the code parameters (particularly, dimension, and minimum distance, etc.) over the parameters obtained by the simple canonical Gray map. The structural properties of constacyclic codes over the ring \(R_{k,m}\) are shown. These codes are later used to obtain new \(q\)-ary quantum codes by employing the known structure of \(\gamma\)-constacyclic codes.
scientific article; zbMATH DE number 7461985
scientific article