Construction and decoding of BCH codes over finite commutative rings

Abstract

BCfi wdes over arbitrary finite ~~~nirnut:!tiv~ rings with identity arc drip& in LCF~S of their locator vector, The derivation is hased on the factorization of .I-'-I over the unit ring of an ~tppropr~lt~ extension of the finite rin g. We prcscnt an ~~~ci~nt,d~~iu~ procedure, based on the modified Berlekamp Massey ;li~~~rithrn. for that codes. The code construction and the decoding proccdurcs arc very similar to the BCH codes over finite integer rings. 43 1999 Ekxvier Scicncc Inc. Ail rights rwrwd. t f MS ~icl.Ev~~~~t~~it, 94BM: 94?35 Linear codes over rings have recently r&xd a great interest for their new role in algebraic coding theory and for their successful application in combined *Corr~s~ndin~ author. E-m& andr~detrt:mat.ithiI~~.unusy.hr, ' This work has bea supertax by Fun&~h dc Amp<> it Pcsquisu do Estudo de So Puub, FAPESP. under Grant No. W4?20-8. and by Consulho Nucion:*t ds Dewnroluimento Cicntilico c Tccnoiciyiw, CNPq. under Grunt No. XHJICdXS-0.

Goppa and Srivastava codes over finite rings

“…We make use of the modified Berlekamp-Massey algorithm to find the solutions of Equation (2). The algorithm is iterative, in the sense that the following n − l n equations (called power sums)…”

“…Linear codes over local finite commutative rings with identity have been discussed in papers by Andrade [1], [2], [3] where it was extended the notion of Hamming, Reed-Solomon, BCH and alternant codes over these rings.…”

Alternant and BCH codes over certain rings

“…Viewing many BCM (block coded modulation) schemes as group block codes over groups, in [3] it was shown that group block codes over abelian groups can be studied via linear codes over finite rings. Andrade and Palazzo [4] constructed BCH codes over finite commutative rings with identity. Also, Greferath and Vellbinger [5] have investigated codes over integer residue rings under the aspect of decoding.…”

Substitution Box on Maximal Cyclic Subgroup of Units of a Galois Ring

“…However, the exponential sums over Galois rings and an upper bound for the hybrid sum over the Galois ring are obtained by using maximal cyclic subgroups of the groups of units of these Galois ring one can see in a series of papers Cohen [2] and Shanbhag et al [3]. Further, Andrade and Palazzo [4] gave the construction of BCH codes over the Galois rings by means of maximal cyclic subgroup. In this sequel, Shah et al [5,6] present a sequence of BCH codes using the chain of maximal cyclic subgroups of the chain of groups of units in the chain of finite Galois rings and finite unitary commutative rings.…”

“…Section 3 consists of the algebraic expression of the proposed S-boxes over maximal cyclic subgroups of groups of units of Galois ring extensions GR(4,2) and GR(4,4) of Z 4 . In Section 4, we examine the security of the projected S-box with the majority logic criterion (MLC).…”