Cyclic codes with few weights and Niho exponents

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Abstract

This paper studies the values of the sums Sk(a)=xF2m(-1)Tr(xk+ax),aF2m,where Tr is the trace function on F2m, m=2t and gcd(2m-1,k)=1. We mainly prove that when k2j(mod2t-1), for some j, then Sk(a) takes at least four values when a runs through F2m. This result, and other derived properties, can be viewed in the study of weights of some cyclic codes and of crosscorrelation function of m-sequences.

Keywords

Finite field
Cyclic code
Maximum-length sequence
Niho exponent
Crosscorrelation
Boolean function
Nonlinearity
Balanced codeword

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