Cyclic codes with few weights and Niho exponents
Pascale Charpin
INRIA, projet CODES
Pascale.Charpin@inria.fr
Jour. Comb. Theory Series A.
Volume 108, Issue 2 , November 2004, Pages 247-259.
Abstract
This paper studies the values of the exponential sums
S_k(a), a in GF(2^m), m=2t,
and k is a Niho exponent such that gcd(2^m-1,k)=1.
That is the sum on x of (-1)^{Tr(x^k+ax)}
where Tr is the trace function on GF(2^m)
and k is congruent to a power of 2 modulo
2^t-1.
We mainly prove that S_k(a)
takes at least four values when a runs through GF(2^m).
This result, and other derived properties,
can be viewed in the study of weights of
some cyclic codes
and of crosscorrelation function of some m-sequences.
KEYWORDS
Finite field, cyclic code, maximum-length sequence, Niho exponent,
crosscorrelation,
Boolean function, nonlinearity, balanced codeword.