Dihedral Codes with Prescribed Minimum Distance

On BCH split metacyclic codes

“…In this note, we generalize some of the results established in [8] to all split metacyclic codes. Our approach follows closely that of [8].…”

“…In this note, we generalize some of the results established in [8] to all split metacyclic codes. Our approach follows closely that of [8]. Besides, we give a polynomial relation between generators of a principal G-code and its dual, which provides an analogue of (1) (see Corollary 3).…”

“…For the second point, C is quasi-cyclic of index m if and only if T •m rm ∈ PAut(C), that is, if and only if ϕ(C) is an R-submodule of R m (by the first point). Now we recall the notion of F q -linear cyclic codes (see [8]), which are another generalization of cyclic codes. To this end, let α ∈ F q m be a primitive element of F q m /F q .…”

“…Recall that BCH codes are named after Bose, Chaudhuri, and Hocquenghem (see [11] and [16]) and that they form a class of powerful multiple-error-correcting. In [8], Borello and Jamous investigate such an analogy for some metacyclic codes, namely the dihedral ones, that is, group codes over dihedral groups. For instance, they define "generator polynomials" and then establish a BCH bound for principal dihedral codes.…”

Dimensión de ideales en álgebras de grupo, y códigos de grupo

“…En [18], abordaron el problema de determinar la dimensión de un código de grupo, que es un ideal principal, estudiando el polinomio característico de la representación regular derecha/izquierda de un generador. Recientemente, en [7] se calcularon cotas para la distancia mínima y dimensión en la clase de códigos BCH-diédricos principales introducidos por los autores, lo cual les permitió dar códigos dihedricos con distancia mínima prescrita.…”