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Cyclic codes over $\text{GR}(p^2,m)$ of length $p^k$. (English)
Finite Fields Appl. 14, No. 3, 834-846 (2008).
The authors refine a method of {\it S. T. Dougherty} et al. [Finite Fields Appl. 13, No. 1, 31‒57 (2007; Zbl 1130.94333)] to represent any ideal of $R=\text{GR}(p^e,m)[u]/\langle u^{p^k}-1\rangle$ in terms of $e$ polynomials as generators in the ideal concerned. (Here $\text{GR}(p^e,m)$ denotes the Galois ring of characteristic $p^e$ with $(p^e)^m$ elements.) The refinement provides a unique representation for any such ideal and enables a classification of all ideals of $R$ in the case $e=2$ (and thus of the cyclic codes of length $p^k$ over GR$(p^2,m)$). The duals of these ideals are analyzed and all selfdual ideals are identified when $p$ is odd and for $k\leq 4$ when $p=2$.
Reviewer: Ralph-Hardo Schulz (Berlin)