On Non-binary Quantum BCH Codes

On classical BCH codes and quantum BCH codes

“…Lemma 3 [16] Let C be a cyclic code of length n over 2 q F such that gcd(n,q)=1, and let S be the index set of roots of C. The code [14] Suppose that…”

“…Cohen et al [11] improved estimates on the parameters of quantum codes obtained by Steane and constructed several new families of quantum BCH codes. Li [13] [14] gave the sufficient and necessary condition for a primitive or some particular non-primitive BCH codes containing their duals. Aly, et al [15] generalized Steane's result in various ways, in particular, to narrow-sense (not necessarily primitive) BCH codes over arbitrary finite fields with respect to Euclidean and Hermitian duals.…”

On the Construction of Nonbinary Quantum BCH Codes

“…Applying the Hermitian construction, an [ [13,9,3]] 5 quantum MDS-BCH code is constructed. Similarly, we can also construct an [ [17,13,3]] 4 and an [ [17,9,5]] 4 quantum MDS-BCH code.…”

“…In this context, many classical codes involved in these constructions are Bose-Chaudhuri-Hocquenghem codes [4,5,12]. Interesting works concerning this class of codes were presented in the literature [1,[14][15][16][17]22]. More precisely, the dimension and sufficient condition (in some cases, necessary and sufficient condition) for dual (Euclidean and Hermitian) containing Bose-Chaudhuri-Hocquenghem codes were investigated.In [1], the authors constructed families of good nonbinary quantum (narrow-sense) codes by showing useful properties of cyclotomic cosets.…”

Quantum Information Security Protocols and Quantum Coding Theory

“…We give five algorithms dealing with the parameters of BCH codes [26]. The first algorithm computes an upper bound of the designed distance of a classical narrow-sense BCH codes which contains its Euclid dual; The second algorithm computes the maximum designed distance when a non-primitive , narrow -sense BCH code can contain its Euclidean dual; The third algorithm computes the dimension of a classical BCH code; The fourth algorithm computes when a non-primitive ,narrow-sense BCH code can contain its Hermitian dual; The fifth algorithm computes when a non-primitive , non-narrow-sense BCH code can contain its Hermitian dual.…”