An Infinite Family of Binary Cyclic Codes With Best Parameters

Binary cyclic codes with parameters
$[n,(n+1)/2, d\geq \sqrt {n}]$
are very interesting, as their minimum distances have a square-root bound. The binary quadratic residue codes and the punctured binary Reed-Muller codes of order
$(m-1)/2$
for odd
$m$
are two infinite families of binary cyclic codes with such parameters. The objective of this paper is to present and analyse an infinite family of binary BCH codes
${\mathcal {C}}(m)$
with parameters
$[2^{m}-1,2^{m-1},d]$
whose minimum distance
$d$
much exceeds the square-root bound when
$m \geq 11$
is a prime. The binary BCH code
${\mathcal {C}}(3)$
is the binary Hamming code and distance-optimal. The binary BCH code
${\mathcal {C}}(5)$
has parameters
$[{31,16,7}]$
and is distance-almost-optimal. The binary BCH code
${\mathcal {C}}(7)$
has parameters
$[{127,64,21}]$
and has the best known parameters. In addition, there is no known
$[2^{m}-1,2^{m-1}]$
binary cyclic code whose minimum distance is better than the minimum distance of this binary BCH code
${\mathcal {C}}(m)$
with parameters
$[2^{m}-1,2^{m-1}]$
for any odd prime
$m$
.