On the Triple-Error-Correcting Cyclic Codes with Zero Set {1, 2^i + 1, 2^j + 1}

We consider a class of 3-error-correcting cyclic codes of length 2^m −1 over the two-element field F2 . The generator polynomial of a code of this class has zeroes α, α^(2^i +1) and α^(2^j +1) , where α is a primitive element of the field F2^m . In short, {1, 2^i + 1, 2^j + 1} refers to the zero set of these codes. Kasami in 1971 and Bracken and Helleseth in 2009, showed that cyclic codes with zeroes {1, 2^l + 1, 2^(3l) + 1} and {1, 2^l + 1, 2^(2l) + 1} respectively are 3-error correcting, where gcd(l , m) = 1. We present a sufficient condition so that the zero set {1, 2^l + 1, 2^(pl) + 1}, gcd( l, m) = 1 gives a 3-error-correcting cyclic code. The question for p > 3 is open. In addition, we determine all the 3-error-correcting cyclic codes in the class {1, 2^i + 1, 2^j + 1} for m < 20. We investigate their weight distribution via their duals and observe that they have the same weight distribution as 3-error-correcting BCH codes for m < 14. Further our experiment shows that these codes are not equivalent to the 3-error-correcting BCH code in general. We also study the Schaub algorithm which determines a lower bound of the minimum distance of a cyclic code. We introduce a pruning strategy to improve the Schaub algorithm. Finally we study the cryptographic property of a Boolean function, called spectral immunity which is directly related to the minimum distance of cyclic codes over F2m . We apply the improved Schaub algorithm in order to find a lower bound of the spectral immunity of a Boolean function related to the zero set {1, 2^i + 1, 2^j + 1}