The Dimension and Bose Distance of Certain Primitive BCH Codes
Run Zheng, Nung-Sing Sze, Zejun Huang · IEEE Transactions on Information Theory · 2025
Bose-Ray-Chaudhuri-Hocquenghem (BCH) codes are a significant class of cyclic codes that play an important role in both theoretical research and practical applications. Their strong error-correcting abilities and efficient encoding and decoding methods make BCH codes widely applicable in various areas, including communication systems, data storage devices, and consumer electronics. Although BCH codes have been extensively studied, the parameters of BCH codes are not known in general. Letqbe a prime power andmbe a positive integer. Denote byC(q,m,δ))the narrow-sense primitive BCH code with lengthqm− 1 and designed distance δ. As of now, the dimensions ofC(q,m,δ)are fully understood only form≤ 2. Form≥ 4, the dimensions ofC(q,m,δ)are known only for the range 2 ≤ δ ≤q⌊(m+1)/2⌋+1and for a limited number of special cases. In this paper, we determined the dimension and Bose distance ofC(q,m,δ)form≥ 4 and δ ∈ [2,q⌊(2m−1)/3⌋+1]. Additionally, we have also extended our results to primitive BCH codes that are not necessarily narrow-sense.