Original research
Construction and decoding of BCH codes over chain of commutative rings
1 Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Pakistan
2 Department of Mathematics, São Paulo State University, São José do Rio Preto, São Paulo, 15054-000, Brazil
Mathematical Sciences 2012, 6:51 doi:10.1186/2251-7456-6-51
Published: 12 October 2012Abstract
In this paper, we present a new construction and decoding of BCH codes over certain
rings. Thus, for a nonnegative integer t, let
be a chain of unitary commutative rings, where each
is constructed by the direct product of appropriate Galois rings, and its projection
to the fields is
(another chain of unitary commutative rings), where each
is made by the direct product of corresponding residue fields of given Galois rings.
Also,
and
are the groups of units of
and
, respectively. This correspondence presents a construction technique of generator
polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing
entries from
and
for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and
error correction capability; however, the proposed contribution offers a choice to
opt a worthy BCH code concerning code rate and error correction capability. In the
second phase, we extend the modified Berlekamp-Massey algorithm for the above chains
of unitary commutative local rings in such a way that the error will be corrected
of the sequences of codewords from the sequences of BCH codes at once. This process
is not much different than the original one, but it deals a sequence of codewords
from the sequence of codes over the chain of Galois rings.
