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A family of MDS abelian group codes. (English) Zbl 1339.94097

Let \(\mathbb{F}\) be a finite field and \(G\) and abelian group. An ideal in the group algebra \(\mathbb{F}G\) can be viewed as a code over \(\mathbb{F}\). In this article the authors study the case in which \(G=\langle h | h^p=1 \rangle\times\langle k | k^q=1 \rangle\), where \(p\) and \(q\) are two distinct primes, so that \(\mathbb{F}G\) consists of \(pq\) idempotents \(e_j\). An ideal of \(\mathbb{F}G\) is a direct sum \(I_{\beta}=\oplus \mathbb{F}Ge_j\), \(e_j\in\beta\), for some subset \(\beta\). They choose suitable subsets \(\beta\) to construct codes \(I_{\beta}\) with maximum distance. They also determine the minimum distance of such \(I_{\beta}\).

MSC:

94B60 Other types of codes
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94B99 Theory of error-correcting codes and error-detecting codes
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