Binary [n, (n + 1)/2] Cyclic Codes With Good Minimum Distances

The binary quadratic-residue codes and the punctured Reed-Muller codes
${\mathcal {R}}_{2}((m-1)/2, m))$
are two families of binary cyclic codes with parameters
$[n, (n+1)/2, d \geq \sqrt {n}]$
. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length
$2^{m}-1$
and dimension near
$2^{m-1}$
with good minimum distances. When
$m \geq 3$
is odd, the codes become a family of duadic codes with parameters
$[2^{m}-1, 2^{m-1}, d]$
, where
$d \geq 2^{(m-1)/2}+1$
if
$m \equiv 3 \pmod {4}$
and
$d \geq 2^{(m-1)/2}+3$
if
$m \equiv 1 \pmod {4}$
. The two families of binary cyclic codes contain some optimal binary cyclic codes.