Abstract
In this paper, cyclic codes over the Galois ring ${\rm GR}({p^2},s)$ are studied. The main result is the characterization and enumeration of Hermitian self-dual cyclic codes of length $p^a$ over ${\rm GR}({p^2},s)$. Combining with some known results and the standard Discrete Fourier Transform decomposition, we arrive at the characterization and enumeration of Euclidean self-dual cyclic codes of any length over ${\rm GR}({p^2},s)$.
Highlights
Cyclic and self-dual codes over finite fields have been extensively studied for both theoretical and practical reasons
We focus on the characterization and enumeration of Hermitian self-dual cyclic codes of length pa over GR(p2, s) and their application to the enumeration of Euclidean self-dual cyclic codes of any length over GR(p2, s)
We discovered that [2, Lemma 7, Lemma 9, and Corollary 2] and [4, Lemma 5.2, Corollary 5.4, Proposition 5.8, Corollary 5.9, and Section 6] contain some errors
Summary
Cyclic and self-dual codes over finite fields have been extensively studied for both theoretical and practical reasons (see [6], [11], and references therein). After that the concepts of cyclic and self-dual codes have been extended and studied over the ring Z4 (see [1], [2], and [4]). In [2], the structure of cyclic a codes of oddly even length (2m, where m is odd) over Z4 has been studied via the Discrete Fourier Transform decomposition. This idea has been extended to the case of all even lengths in [4]. A remarkable structural decomposition of cyclic codes over Z4 are given in [2] and [4]. [2, Lemma 7, Lemma 9, and Corollary 2] and [4, Lemma 5.2, Corollary 5.4, Proposition 5.8, Corollary 5.9, and Section 6] are not correct
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