Another Infinite Family of Binary Cyclic Codes With Best Parameters Known

Cyclic codes are important in theory, as they are closely related to a number of areas of mathematics. Cyclic codes are also important in practice, as they have efficient encoding and decoding algorithms. An infinite family of cyclic codes over <inline-formula> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> is said to have linearly-best-known parameters if for any <inline-formula> <tex-math notation="LaTeX">$[n, k, d]$ </tex-math></inline-formula> code <inline-formula> <tex-math notation="LaTeX">${\mathcal {C}}$ </tex-math></inline-formula> in this family, there is no known <inline-formula> <tex-math notation="LaTeX">$[n, k, d']$ </tex-math></inline-formula> linear code over <inline-formula> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$d' > d$ </tex-math></inline-formula>. An infinite family of cyclic codes over <inline-formula> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> is said to have cyclicly-best-known parameters if for any <inline-formula> <tex-math notation="LaTeX">$[n, k, d]$ </tex-math></inline-formula> code <inline-formula> <tex-math notation="LaTeX">${\mathcal {C}}$ </tex-math></inline-formula> in this family, there is no known <inline-formula> <tex-math notation="LaTeX">$[n, k, d']$ </tex-math></inline-formula> cyclic code over <inline-formula> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$d' > d$ </tex-math></inline-formula>. It is very rare to see an infinite family of binary cyclic codes with cyclicly-best-known parameters whose duals codes have also cyclicly-best-known parameters. The objective of this paper is to study such family of binary cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$2^{m}-1$ </tex-math></inline-formula> and dimension <inline-formula> <tex-math notation="LaTeX">$2^{m}-1-m(m-1)/2$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">${\mathcal {C}}_{(2,m,2)}$ </tex-math></inline-formula>, and their dual codes <inline-formula> <tex-math notation="LaTeX">${\mathcal {C}}_{(2,m,2)}^{\perp} $ </tex-math></inline-formula>. The weight distribution of <inline-formula> <tex-math notation="LaTeX">${\mathcal {C}}_{(2,m,2)}^{\perp} $ </tex-math></inline-formula> is settled and the parameters of <inline-formula> <tex-math notation="LaTeX">${\mathcal {C}}_{(2,m,2)}$ </tex-math></inline-formula> are investigated in this paper. A larger family of binary cyclic codes <inline-formula> <tex-math notation="LaTeX">${\mathcal {C}}_{(2,m,r)}$ </tex-math></inline-formula> and their duals are also constructed and studied in this paper, where <inline-formula> <tex-math notation="LaTeX">$0 \leq r \leq m-1$ </tex-math></inline-formula>.