Abstract
Quasicyclic codes of length n = mℓ and index ℓ over the finite field F q are linear codes invariant under cyclic shifts by ℓ places. They are shown to be isomorphic to the F q[x]/〈x m−1〉-submodules of F q ℓ[x]/〈x m−1〉 where the defining property in this setting is closure under multiplication by x with reduction modulo x m−1. Using this representation, the dimension of a 1-generator code can be determined straightforwardly from the chosen generator, and improved lower bounds on minimum distance are developed. A special case of multi-generator codes, for which the dimension can be algebraically recovered from the generating set is described. Every possible dimension of a quasicyclic code can be obtained in some such special form. Lower bounds on minimum distance are also given for all multi-generator quasicyclic codes.
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Lally, K. (2003). Quasicyclic Codes of Index ℓ over F q Viewed as F q[x]-Submodules of F q ℓ[x]/〈x m−1〉. In: Fossorier, M., Høholdt, T., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2003. Lecture Notes in Computer Science, vol 2643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44828-4_26
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DOI: https://doi.org/10.1007/3-540-44828-4_26
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