Abstract

GARCIA-RAMIREZ, Angel R.; LOPEZ-ANDRADE, Carlos A.  and  VILLA-HERNANDEZ, David. Gray images of constacyclic codes over Galois rings R of nilpotency index 3. Integración - UIS [online]. 2021, vol.39, n.1, pp.57-78.  Epub Feb 28, 2021. ISSN 0120-419X.  https://doi.org/10.18273/revint.v39n1-2021005.

We will state necessary and sufficient conditions for the image under the Gray map of a R-constacyclic code to bepm-quasi-cyclic code. We study the Witt vectors to get a way to operate the µ-reduction of padic components of the elements of the Galois rings of nilpotency index 3, R = GR (p3 , m). We analyze Galois rings, its mostly relevant properties, and we focus in the p-adic representation of their elements. Later on, we examine construction of the Witt vectors rings and its operations, in particular, we get explicit expressions for operations of addition and product of the elements in the truncated Witt vectors ring of length 3, W3 (pm). Finally, we will use these operations and an isomorphism between GR (p3 , m) and W3 (pm) to get a way to operate the µ-reductions described above.

Keywords : Witt’s vectors ring; Galois rings; p-adic representation; constacyclic codes; quasi-cyclic codes.

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