3 research outputs found
Constacyclic codes of length over the Galois ring
For prime , represents the Galois ring of order and
characterise , where is any positive integer. In this article, we study
the Type (1) -constacyclic codes of length over the ring
, where , are
nonzero elements and . In first case, when is a
square, we show that any ideal of
is the direct sum of the ideals of
and
. In second, when
is not a square, we show that is a chain
ring whose ideals are , for where .
Also, we prove the dual of the above code is and
present the necessary and sufficient condition for these codes to be
self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman
(RT) distance, Hamming distance and weight distribution of Type (1)
-constacyclic codes of length are obtained when is
not a square.Comment: This article has 18 pages and ready to submit in a journa
Imágenes de Gray de códigos consta-cÃclicos sobre anillos de Galois R de Ãndice de nilpotencia 3.
We will state necessary and sufficient conditions for the image under the Gray map of a R-constacyclic code to be Fpm-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(Fpm). Finally, we will use these operations and an isomorphism between GR(p3, m) and W3(Fpm) to get a way to operate the µ-reductions described above.Estableceremos condiciones necesarias y suficientes para que la imagen bajo la función de Gray de un R-código consta-cÃclico sea un Fpmcódigo cuasi-cÃclico. Estudiamos el anillo de vectores de Witt para obtener una manera de operar las µ-reducciones de las componentes p-ádicas de los elementos de los anillos de Galois de Ãndice de nilpotencia 3, R = GR(p3, m). Analizamos a los anillos de Galois, sus propiedades más relevantes, y en particular la representación p-ádica de sus elementos. Más adelante, examinamos la construcción del anillo de vectores de Witt y sus operaciones, en particular, obtenemos expresiones explÃcitas para las operaciones de suma y producto de los elementos en el anillo truncado de vectores de Witt de longitud 3, W3(Fpm). Finalmente, utilizamos las operaciones de éstos últimos y un isomorfismo entre GR(p3, m) y W3(Fpm) para operar las µ-reducciones antes descritas