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International Conference on Machine Learning and Cybernetics

Constructions of quantum BCH codes based on finite fields of odd characteristic

You Gao, Yu-Ting Xiao, Ya-Yuan Tao

Published Jul 1, 2010• 0 citations
Publisher

Abstract

Let Fqbe finite field with q elements, where q is a power of an odd prime p. Sufficient and necessary conditions for the self orthogonality of two classes of classical q-ary non-primitive BCH codes over Fqof length n = qm-1 and n = qr-1/qm-1 are given, respectively. These conditions allow us to derive the parameters of four series of g-ary quantum BCH codes as a function of their design parameters based on some construction methods.

DOI
10.1109/icmlc.2010.5580590

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Let Fqbe finite field with q elements, where q is a power of an odd prime p. Sufficient and necessary conditions for the self orthogonality of two classes of classical q-ary non-primitive BCH codes over Fqof length n = qm-1 and n = qr-1/qm-1 are given, respectively. These conditions allow us to derive the parameters of four series of g-ary quantum BCH codes as a function of their design parameters based on some construction methods.
Let q ≥ 3 be odd prime power. Maximal designed distances of non-primitive Hermitian dual containing q2- ary BCH codes of length n = (q4-1)/2 and 2(q2-1) are determined. For each given n, a group of non-narrow-sense Hermitian dual containing BCH codes are constructed and their dimensions are computed. Using these non-narrow-sense BCH codes, many new quantum codes are constructed. Our maximal designed distances improve upon the known results in the literature and many of new quantum BCH codes are better than those constructed from narrow-sense BCH codes.
Maximal designed distances for nonbinary narrow-sense quantum Bose-Chaudhuri-Hocquenghem (BCH) codes of length n=q4−1rn=\frac{q^4-1}{r}n=rq4−1​ and new constructions for them are given, where q is an odd prime power. These constructions are capable of designing quantum BCH codes with new parameters. Furthermore, some codes obtained here have better parameters than those constructed by other known constructions.
An attractive feature of BCH codes is that one can infer valuable information from their design parameters (length, size of the finite field, and designed distance), such as bounds on the minimum distance and dimension of the code. In this paper, it is shown that one can also deduce from the design parameters whether or not a primitive, narrow-sense BCH contains its Euclidean or Hermitian dual code. This information is invaluable in the construction of quantum BCH codes. A new proof is provided for the dimension of BCH codes with small designed distance, and simple bounds on the minimum distance of such codes and their duals are derived as a consequence. These results allow us to derive the parameters of two families of primitive quantum BCH codes as a function of their design parameters