Codes over affine algebras with a finite commutative chain coefficient ring

Abstract

We consider codes defined over an affine algebra A = R[X1, . . . , Xr]/ t1(X1), . . . , tr(Xr) , where ti(Xi) is a monic univariate polynomial over a finite commutative chain ring R. Namely, we study the A−submodules of A l (l ∈ N). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. Some codes over Frobenius local rings that are not chain rings are also of this type. A canonical generator matrix for these codes is introduced with the help of the Canonical Generating System. Duality of the codes is also considered.

Cyclic codes over a non-chain ring $R_{e,q}$ and their application to LCD codes

“…In this section, we determine the structure of cyclic codes of length n over R e,q and use these results to obtain LCD codes in the subsequent section. Note that in [27] the authors considered this type of codes as codes over an affine algebras with a finite commutative chain coefficient ring where the affine algebra is R e,q .…”

“…As we can see in Tables 2 and 4, the codes of parameters [16,11,4], [20,15,4], [24,19,4], [32,27,4], [12,8,3], [12,7,4], [16,12,3], [26,22,3]…”

Linear Complementary Pair Of Group Codes over Finite Chain Rings

“…Note that an explicit polynomial description of the dual code of a code C in R[G] in the nonrepeated root case and the repeated root case can be found in [17] and [16] respectively. (i…”

On polycyclic codes over a finite chain ring

“…Also in [4] some classes of constacyclic codes over finite chain rings were characterized via cyclotomic cosets as contraction of some cyclic codes. In [11], the authors studied the algebraic structure of a class of linear codes over finite chain rings containing the polycyclic codes.…”