On Bose distance of a class of BCH codes with two types of designed distances

BCH codes are an interesting class of cyclic codes with good error-correcting capability and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Let 𝔽_q be the finite field of size q and n=q^m-1 , where m is a positive integer. Let 𝒞_(q, m, δ ) be the primitive narrow-sense BCH codes of length n over 𝔽_q with designed distance δ . Denote s = m - t , r = m s and λ = ⌊ t/s ⌋ . In this paper, we mainly investigate the dimensions and Bose distances of the codes 𝒞_(q, m, δ ) with designed distance of the following two types: This extensively extends the results on Bose distance in Ding et al (IEEE Trans Inf Theory 61(5):2351–2356, 2015). Moreover, the parameters of the hulls of the BCH code 𝒞_(q, m, q^t) are studied in some cases.