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=mmods 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: δ=q^t+h, ⌈m2⌉≤t<m, 0≤h<q^s+∑i=1λ-1q^r+is; δ=q^t-h, ⌈m2⌉<t<m, 0≤h<(q-1)∑i=1sqⁱ. 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,qt) are studied in some cases.