On the triple-error-correcting cyclic codes with zero set {1, 2i + 1, 2i + 1}

• Authors: Vincent Herbert;Sumanta Sarkar
• Venue: IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
• Year: 2011

We consider a class of 3-error-correcting cyclic codes of length 2m −1 over the two-element field $\mathbb{F}_2$ . The generator polynomial of a code of this class has zeroes $\alpha, \alpha^{2^i+1}$ and $\alpha^{2^j+1}$ , where α is a primitive element of the field ${\mathbb{F}_{2^m}}$ . In short, {1, 2i +1, 2j +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, 2ℓ+1, 23ℓ+1} and {1, 2ℓ+1, 22ℓ+1} respectively are 3-error correcting, where $\gcd(\ell,m) = 1$ . We present a sufficient condition so that the zero set {1, 2ℓ+1, 2p ℓ+1}, $\gcd(\ell,m)=1$ 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, 2i +1, 2j +1} for m m ${\mathbb{F}_{2^m}}$ . 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, 2i +1, 2j +1}.