Triple-error-correcting BCH-like codes

The binary primitive triple-error-correcting BCH code is a cyclic code of minimum distance d = 7 with generator polynomial having zeros alpha, alpha
³
and alpha
⁵
where alpha is a primitive (2
ⁿ
- 1)-root of unity. The zero set of the code is said to be {1, 3, 5}. In the 1970's Kasami showed that one can construct similar triple-error-correcting codes using zero sets consisting of different triples than the BCH codes. Furthermore, in 2000 Chang et. al. found new triples leading to triple-error-correcting codes. In this paper a new such triple is presented. In addition a new method is presented that may be of interest in finding further such triples. The method is illustrated by giving a new and simpler proof of one of the known Kasami triples {1, 2
^k
+ 1, 2
^3k
+ 1} where n is odd and gcd(k, n) = 1 as well as to find the new triple given by {1, 2
^k
+ 1, 2
^2k
+ 1} for any n where gcd(k, n) = 1.