A BCH Code and a Sequence of Cyclic Codes

This study establishes that for a given binary BCH code C0 n of length n generated by a polynomial g(x) ∈ F2[x] of degree r there exists a family of binary cyclic codes {Cm 2m−1(n+1)n}m≥1 such that for each m ≥ 1, the binary cyclic code Cm 2m−1(n+1)n has length 2 m−1(n + 1)n and is generated by a generalized polynomial g(x 1 2m ) ∈ F2[x, 1 2m Z≥0] of degree 2mr. Furthermore, C0 n is embedded in C m 2m−1(n+1)n and C m 2m−1(n+1)n is embedded in C 2m(n+1)n for each m ≥ 1. By a newly proposed algorithm, codewords of the binary BCH code C0 n can be transmitted with high code rate and decoded by the decoder of any member of the family {Cm 2m−1(n+1)n}m≥1 of binary cyclic codes, having the same code rate. Mathematics Subject Classification: 11T71, 14A50, 94A15


Introduction
In [4] Cazaran and Kelarev introduce the necessary and sufficient conditions for the ideal to be a principal ideal and describe all finite principal ideal rings Z m [x 1 , x 2 , • • • , x n ]/I, where I is generated by univariate polynomials.Moreover, in [5], they obtained conditions for certain rings to be finite commutative principal ideal rings.However, the extension of a BCH code embedded in a semigroup ring F[S], where F is a field and S is a finite semigroup, introduced by Cazaran et al. [6], in which an algorithm is considered for computing the weights of extensions for codes embedded in F[S] as ideals.Valuable information related to several ring constructions and concerning polynomial codes was given by Kelarev [8] and [9].Whereas, in [10] and [11], Kelarev discuss the concerning extensions of BCH codes in several ring constructions, where the results can also be considered as particular cases of semigroup rings of particular nature.Andrade and Palazzo [1] elaborated the cyclic, BCH, alternant, Goppa and Srivastava codes over finite rings, which are in real meanings constructed through a polynomial ring in one indeterminate with a finite coefficient ring.Shah et al. [12] and [13], instead of a polynomial ring, the construction methodology of cyclic, BCH, alternant, Goppa, and Srivastava codes over a finite ring is used through a semigroup ring, where the results of [1] are improved in such a way that in the place of cancellative torsion free additive monoid Z ≥0 of non-negative integers, the cancellative torsion free additive monoids 1  2 Z ≥0 and 1 2 2 Z ≥0 are taken, respectively.This converts the whole construction of a finite quotient ring of a polynomial ring into a finite quotient ring of monoid rings of particular nature.In [12] and [13], R is considered as a finite unitary commutative ring for the quotient rings R[x; 1  2 Z ≥0 ]/((x 2 ) 2 2 n − 1), respectively.However, in [2] Andrade et al. describe the decoding principle based on modified Berlekamp-Massey algorithm for BCH, alternant and Goppa codes constructed through monoid rings R[x; 1  2 Z ≥0 ].The existence of an ((n + 1) where k is a positive integer, corresponding to a (n, n − r) binary cyclic code established in [14] through the monoid ring F 2 [x; 1  3 k Z ≥0 ].Furthermore, in [14] a decoding procedure for an (n, n−r) binary cyclic code by an ((n+1) 3 k −1, (n+ 1) 3 k − 1 − 3 k r) binary cyclic code is also given, which provides an improvement in the code rate and error corrections capabilities.
Provoked by [14] we initiate the inquiry in support to binary BCH codes alike binary cyclic codes however we observed that; for a binary BCH code of length n = 2 s − 1 generated by r degree polynomial g(x) ∈ F 2 [x] it is not possible to construct a binary BCH code of length 2 m−1 (n + 1)n generated by 2 m r degree generalized polynomial g(x Though, in this study, we instituted that corresponding to an (n, . Furthermore, we propose an algorithm which enables in decoding of a received vector of binary BCH code C 0 n of length n through the decoding of corresponding generalized received vector of any member of the family {C m 2 m−1 (n+1)n } m≥1 of binary cyclic codes.[7,Theorem 8.4].Corresponding to principal ideal (f (x and it is a field if and only if f (x Clearly, it follows the following proposition.Proposition 2.1 Let g(x) ∈ F 2 [x, Z ≥0 ] be an r degree polynomial.If n = 2 s − 1, where s is a positive integer and g(x) divides x n − 1, then the generalized polynomial g(x In particular, take the product x ; indeed, corresponding to the generalized polynomials c(x ) in F Thus, the isomorphism between the vector spaces is a linear code.As already agreed, we recognize every vector c in F 2 m−1 (n+1)n with the polynomial c(x are now referred as codewords or code (generalized) polynomials.By use of the techniques of [14], the following results can easily be established for 2 m−1 (n + 1)n instead of (n + 1) Note that (p(x represents the principal ideal generated by the polynomial p(x Theorem 2.4 [14] For 1. there exists a unique monic polynomial g(x , where m ≥ 1, is the ideal generated by p(x By Theorem 2.4, if follows that only ideals in the ring )n are linear codes which are generated by the factors of (x Thus we can obtain all cyclic codes of length 2 m−1 (n+1)n over F 2 if we find all factors of (x In the case of trivial factors, we get trivial codes.If g(x ) is monic and divides (x

Relationship of a BCH code and a cyclic code
Let C 0 n be an (n, n − r) binary BCH code based on the positive integers c, δ 1 , q = 2 and n such that 2 ≤ δ 1 ≤ n with gcd(n, 2) = 1 and n = 2 s − 1, where s ∈ Z + .Consequently, the binary BCH code C 0 n has generator polynomial ) is a principal ideal in the factor ring F 2 [x] n .As it is established in Proposition 2.1 that the generalized polynomial g(x . By third isomorphism theorem for rings and ) is the generator polynomial for the cyclic code C m+1 2 m (n+1)n in the monoid ring ).The above discussion shapes the following theorem.
of degree r, then 1. there exists a family {C m 2 m−1 (n+1)n } m≥1 of binary cyclic codes such that for each m ≥ 1 C m 2 m−1 (n+1)n has length 2 m−1 (n + 1)n, generated by the generalized polynomial g(x

the binary BCH code
Is it possible for a binary BCH code C 0 n = (g(x)) that there is a binary BCH code C m 2 m−1 (n+1)n generated by polynomial g(x The answer is no, indeed, as we know that generator polynomial of a binary BCH code is the least common multiple of irreducible polynomials over F 2 .For instance, if g(x) = r i=0 g i x i is the generator polynomial of the binary BCH code C 0 n , then g(x ) is not qualified for a generator of a binary BCH code.

General decoding principle
McEliece, Berlekamp and Van Tilborg [3] proved that the maximum likelihood decoding is an NP-hard problem for general linear codes.Though by the principle of maximum likelihood decoding we obtain a codeword after decoding which is closest to the received vector while the errors are corrected.We use the decoding procedure which follows the same principle.Now, we interpret the decoding terminology for a 2 m 0 −1 (n + 1)n length binary cyclic code C m 0 2 m 0 −1 (n+1)n from the family {C m 2 m−1 (n+1)n } m≥1 of binary cyclic codes.Let parity check matrix of a binary cyclic code C m 0 2 m 0 −1 (n+1)n be H.If a vector b is received, then we obtain the syndrome vector for b as S(b) = bH T .In this way, we calculate syndrome table which is useful in finding the error vector e such that S(b) = S(e).So the decoding of the received vector b has done as the transmitted vector a = b − e.
The general principle of decoding is; choose the codeword which is closest to the received vector.For this determination, we make a look-up table that gives the nearest codeword for every possible received vector.The algebraic structure of a linear code as a subspace offers a suitable method for making such a table.As is a subgroup of the additive group F . Recall that for every a ∈ F These cosets form a partition of the space F . Hence is the disjoint union of distinct cosets.Let y be any vector in F , and suppose x ∈ C m 0 2 m 0 −1 (n+1)n is the codeword nearest to y.Now x lies in the coset y y, c), i.e., w(y − x) ≤ w(y − c).Hence, y − x is the vector of least weight in the coset containing y. Writing e = y − x, we have x = y − e.Thus the following theorem is obtained.
be a cyclic code.Given a vector y ∈ F , the codeword x nearest to y is given by x = y − e, where e is the vector of least weight in the coset containing y.If the coset containing y has more than one vector of least weight, then there are more than one codewords nearest to y. Definition 4.2 Let C m 0 2 m 0 −1 (n+1)n be a linear code in F . The coset leader of a given coset of C m 0 2 m 0 −1 (n+1)n is defined to be the vector with the least weight in the coset. 2 m 0 −1 (n+1)n be an (2 m 0 −1 (n + 1)n, 2 m 0 −1 (n + 1)n − 2 m 0 r) code over F 2 , and let H be a parity-check matrix of By Theorem 4.3, it follows that S(y) = 0 if and only if y ∈ C m 0 2 m 0 −1 (n+1)n .For y, y / ∈ F 2 m 0 −1 (n+1)n , S(y) = S(y ) holds if and only if (y−y )H T = 0, that is, y − y ∈ C m 0 2 m 0 −1 (n+1)n .Hence two vectors have the same syndrome if and only if they lie in the same coset of C m 0 2 m 0 −1 (n+1)n .Thus there is a one-to-one correspondence between the cosets of C m 0 2 m 0 −1 (n+1)n and the syndromes.A table with two columns showing the coset leader e i and the corresponding syndromes S(e i ) is called the syndrome table.To decode a received vector y, we compute its syndrome S(y) and then look at the table to find the coset leader e for which S(e) = S(y).Then y is decoded as x = y − e.The syndromes are given by S(e i ), where e i for i = 1, 2, • • • , 2 2 m 0 r are the coset leaders, F = F 2 and S(e i ) = e i H T , for i = 1, 2, Consider a binary BCH code C 0 n based on the positive integers c, δ, q = 2 and n such that 2 ≤ δ ≤ n with n = 2 s − 1, where s is a positive integer.Let ζ be a primitive n th root of unity in Assume that for a fixed m = m 0 , C m 2 m−1 (n+1)n is a binary cyclic code of length 2 m−1 (n + 1)n = n ) with minimum distance d and with generator generalized polynomial g(x 1 2 m ) from the corresponding family {C m 2 m−1 (n+1)n } m≥1 of binary cyclic codes, which has the check generalized polynomial h(x m ).Thus, the matrix H is given by ⎡ , the generalized polynomial is a(x ) ∈ F Step 6: Decode b as b − e = a .
Step 7: The corresponding corrected codeword polynomial a(x) in binary BCH code C 0 n is obtained.

1 2 m 1 2
)) is the ideal generated by p(x m ), then p(x 1 2 m ) is the generator generalized polynomial of C m 2 m−1 (n+1)n if and only if p(x 1 2 m
and it behave like an indeterminate x in F 2 [x].For instance for a torsion free cancellative monoid S the monoid ring F 2 [x; S] is a Euclidean domain if F 2 is a field and S ∼ = Z or S ∼ = Z ≥0