Binary [n, (n + 1)/2] cyclic codes with good minimum distances

-The binary quadratic -residue codes and the punctured Reed -Muller codes R.2 ((ru - 1)/2, m)) are two families of binary cyclic codes with parameters [n, (n +1)/2, d > TL]. 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'" - 1 and dimension near 2m-1 with good minimum distances. When in > 3 is odd, the codes become a family of duadic codes with parameters [2m - 1, 2m-1, d], where d > 2("'-1)/2 + 1 if nt 3 (mod 4) and d > 2(m-1)/2 + 3 if rn = 1 (mod 4). The two families of binary cyclic codes contain some optimal binary cyclic codes.