Another Infinite Family of Binary Cyclic Codes With Best Parameters Known

Abstract

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 ${\mathrm {GF}}(q)$ is said to have linearly-best-known parameters if for any $[n, k, d]$ code ${\mathcal {C}}$ in this family, there is no known $[n, k, d']$ linear code over ${\mathrm {GF}}(q)$ such that $d' > d$ . An infinite family of cyclic codes over ${\mathrm {GF}}(q)$ is said to have cyclicly-best-known parameters if for any $[n, k, d]$ code ${\mathcal {C}}$ in this family, there is no known $[n, k, d']$ cyclic code over ${\mathrm {GF}}(q)$ such that $d' > d$ . 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 $2^{m}-1$ and dimension $2^{m}-1-m(m-1)/2$ , denoted by ${\mathcal {C}}_{(2,m,2)}$ , and their dual codes ${\mathcal {C}}_{(2,m,2)}^{\perp} $ . The weight distribution of ${\mathcal {C}}_{(2,m,2)}^{\perp} $ is settled and the parameters of ${\mathcal {C}}_{(2,m,2)}$ are investigated in this paper. A larger family of binary cyclic codes ${\mathcal {C}}_{(2,m,r)}$ and their duals are also constructed and studied in this paper, where $0 \leq r \leq m-1$ .