On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes. (English) Zbl 1518.94138
Summary: In this paper, we first discuss linear codes over \(R\) and present the decomposition structure of linear codes over the mixed alphabet \(\mathbb{F}_q R\), where \(R=\mathbb{F}_q +u\mathbb{F}_q +v\mathbb{F}_q +uv\mathbb{F}_q\), with \(u^2 = 1, v^2 = 1, uv = vu\) and \(q = p^m\) for odd prime \(p\), positive integer \(m\). Let \(\theta\) be an automorphism on \(\mathbb{F}_q\). Extending \(\theta\) to \(\Theta\) over \(R\), we study skew \((\theta, \Theta)\)-\((\lambda,\Gamma)\)-constacyclic codes over \(\mathbb{F}_q R\), where \(\lambda\) and \(\Gamma\) are units in \(\mathbb{F}_q\) and \(R\), respectively. We also show that, the dual of a skew \((\theta, \Theta)\)-\((\lambda, \Gamma)\)-constacyclic code over \(\mathbb{F}_q R\) is a skew \((\theta, \Theta)\)-\((\lambda^{-1},\Gamma^{-1})\)-constacyclic code over \(\mathbb{F}_q R\). We classify some self-dual skew \((\theta, \Theta)\)-\((\lambda, \Gamma)\)-constacyclic codes using the possible values of units of \(R\). Also using suitable values of \(\lambda, \theta, \Gamma\) and \(\Theta\), we present the structure of other linear codes over \(\mathbb{F}_q R\). We construct a Gray map over \(\mathbb{F}_q R\) and study the Gray images of skew \((\theta, \Theta)\)-\((\lambda,\Gamma)\)-constacyclic codes over \(\mathbb{F}_q R\). As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew \(\theta\)-cyclic codes than from cyclic codes over \(\mathbb{F}_q\).
MSC:
| 94B15 | Cyclic codes |
| 94B60 | Other types of codes |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 81P70 | Quantum coding (general) |
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