Codes through Monoid Rings and Encoding

Cazaran and Kelarev [2] have given necessary and sufficient conditions for an ideal to be the principal; further they described all finite factor rings Zm[X1, · · · , Xn]/I, where I is an ideal generated by an univariate polynomial, which are commutative principal ideal rings. But in [3], Cazaran and Kelarev characterize the certain finite commutative rings as a principal ideal rings. Though, the extension of a BCH code C embedded in a semigroup ring F [T ], where T is a finite semigroup, was considered in 2006 by Cazaran et. all [4], where an algorithm was given for computing the weights of extensions for these codes embedded in semigroup rings as ideals. Kelarev [5] provides the information relating various ring constructions and about polynomial codes, where in Sections 9.1 and 9.2 which are very closely related to semigroup rings, devoted for error-correcting codes in ring constructions . Section 9.1 is dealing error-correcting cyclic codes of length n which are ideals in group ring F [G] with F a field and G a finite torsion group of order n. Another work concerning extensions of BCH codes in various ring constructions has been given by Kelarev in [6] and [7], where the results can also be considered as the special cases of particular type of semigroup rings. A polynomial ring A[X] is initially a semigroup ring A[X;S], where S is the cyclic (additive) monoid Z0, the non negative integers. If A ⊆ B is a unitary commutative ring extension, then in the polynomial ring set up, we call an element b ∈ B is integral over A if b satisfies a monic polynomial over A. The study in [8] address the question: Given a rational cyclic monoid S =< ab > , where a, b ∈ Z), which is not a group, will the semigroup ring A[X;S] behave like a polynomial ring? The answer is no, that can be ascertained by setting X a b = Y and treat Y as an indeterminate over A. To establish this the notion of pseudo integrality relative to rational cyclic monoid S is took place, which generalizes integral dependence. A. A. Andrade and R. Palazzo Jr. [9] discussed the cyclic, BCH, alternant, Goppa and Srivastava codes through the polynomial ring B[X;Z0], where B is any finite commutative ring with identity. By the motivation of [8], in [1] we constructed cyclic codes through monoid ring B[X; ab Z0], where b ≥ a with b = a + 1, instead of a polynomial ring B[X] as considered in [9]. Nevertheless we focused only on encoding as [10], whereas the decoding procedure like [11] is required a separate section of discussion. In this study we continue the the procedure adopted in [1] for construction of cyclic codes through monoid ring B[X; ab Z0], where b ≥ a with b = a + 2. The procedure used in this study for constructing linear codes through the monoid ring B[X; ab Z0] is simple like polynomial’s set up and technique adopted here is quite different to the embedding of linear polynomial codes in a semigroup ring or in a group algebra, which has been considered by many authors. For example, in [5], the Sections 9.1 is dealing with error-correcting cyclic codes of length n which are ideals in group ring F [G], whereas G is taken to be a finite torsion group of order n.