The True Minimum Distance of the Antiprimitive BCH Codes With Designed Distance 3

Abstract

Let <inline-formula> <tex-math notation="LaTeX">$\mathcal {C}_{(q,q^{m}+1,3,h)}$ </tex-math></inline-formula> denote the antiprimitive BCH code with designed distance 3. For any <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, we demonstrate that the minimum distance <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$\mathcal {C}_{(q,q^{m}+1,3,h)}$ </tex-math></inline-formula> equals 3 if and only if <inline-formula> <tex-math notation="LaTeX">$\gcd (2h+1,q+1,q^{m}+1)\ne 1$ </tex-math></inline-formula>. For odd <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, we show that <inline-formula> <tex-math notation="LaTeX">$d=4$ </tex-math></inline-formula> if and only if <inline-formula> <tex-math notation="LaTeX">$\gcd (2h+1,q+1)=1$ </tex-math></inline-formula>, thereby fully characterizing the minimum distance in this case. For even <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> or even <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, we establish some sufficient conditions for <inline-formula> <tex-math notation="LaTeX">$d=4$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$d=5$ </tex-math></inline-formula>. Additionally, we investigate the parameters of <inline-formula> <tex-math notation="LaTeX">$\mathcal {C}_{(q,q^{m}+1,3,h)}$ </tex-math></inline-formula> for certain <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula>, and present two infinite families of distance-optimal codes as well as several linear codes with the best known parameters.