An association between primitive and non-primitive BCH codes using monoid rings

Abstract

BCH codes are one of the most important classes of cyclic codes for error correction. In this study, we generalize BCH codes using monoid rings instead of a polynomial ring over the binary field F 2 . We show the existence of a non-primitive binary BCH code C bn of length bn, corresponding to a given length n binary BCH code C n . The value of b is investigated for which the existence of the non-primitive BCH code C bn is assured. It is noticed that the code C n is embedded in the code C bn . Therefore, encoding and decoding of the codes C n and C bn can be done simultaneously. The data transmitted by C n can also be transmitted by C bn . The BCH code C bn has better error correction capability whereas the BCH code C n has better code rate, hence both gains can be achieved at the same time.

Sequences of Primitive and Non-primitive BCH Codes

“…This work extends the work of [3], where the monoid ring F 2 [x; a b j Z 0 ], with b = a+i, 1 ≤ i, j ≤ m, where m is a positive integer, is used. Corresponding to the sequence {F 2 [x; a b j Z 0 ]} j≥1 of monoid rings, we obtain a sequence of non-primitive binary BCH codes {C j b j n } j≥1 , based on a primitive BCH code C n of length n. The non-primitive BCH code C j b j n in the monoid ring F 2 [x; a b j Z 0 ] is of length b j n and generated by a generalized polynomial g(x a b j ) ∈ F 2 [x; a b j Z 0 ] of degree b j r. Similarly, corresponding to a given binary BCH code C n of length n generated by a polynomial g(x a ) ∈ F 2 [x; aZ 0 ] of degree r it is constructed a code C j b j n such that C n is embedded in C j b j n , where the length of the binary BCH code C j b j n is well controlled with better error correction capability.…”

“…Moreover, a link between all these codes are also been developed. Furthermore, in [3], the work of [16] is improved and an association between primitive and non-primitive binary BCH codes is obtained by using the monoid ring F 2 [x; a b Z 0 ], where a, b > 1. It is noticed that the monoid ring F 2 [x; a b Z 0 ] does not contain the polynomial ring F 2 [x] for a > 1.…”

“…The binary bits of v 1 are properly overlapped on the bits of v 2 , in fact, they are repeated three times after a particular pattern. Hence, the generating matrix G 2 of g(x 9 ) contains the generating matrix G 1 of g( x 23 ) such that G 2 = ⊕ 3 1 G 1 .…”

“…[3, Theorem 6] Let C n be a primitive binary BCH code of length n = 2 s −1 generated by a polynomial g(x a ) of degree r in F 2 [x; aZ 0 ].…”