The Moments of a Kloosterman Sum and the Weight Distribution of a Zetterberg-Type Binary Cyclic Code

In this correspondence, we give the moments of a Kloosterman sum over Fq in terms of the frequencies of weights in the binary Zetterberg code of length q+1, which are known by the work of Schoof and van der Vlugt. The method is illustrated by giving explicit formulae for the moments up to the tenth moment. As a corollary the weight distribution of a Zetterberg-type binary cyclic code is obtained

[1]  G. Andrews,et al.  Encyclopedia of Mathematics and its Applications , 1990 .

[2]  Tor Helleseth,et al.  The minimum distance of the duals of binary irreducible cyclic codes , 2002, IEEE Trans. Inf. Theory.

[3]  Marko J. Moisio,et al.  On the moments of Kloosterman sums and fibre products of Kloosterman curves , 2008, Finite Fields Their Appl..

[4]  L. Carlitz.  A note on exponential sums , 1969 .

[5]  Marko J. Moisio,et al.  On the number of rational points on some families of Fermat curves over finite fields , 2007, Finite Fields Their Appl..

[6]  Tor Helleseth,et al.  Niho type cross-correlation functions via dickson polynomials and Kloosterman sums , 2006, IEEE Transactions on Information Theory.

[7]  Tor Helleseth,et al.  Some results about the cross-correlation function between two maximal linear sequences , 1976, Discret. Math..

[8]  Vera Pless,et al.  Power Moment Identities on Weight Distributions in Error Correcting Codes , 1963, Inf. Control..

[9]  L. Carlitz.  Gauss sums over finite fields of order $2^n$ , 1969 .

[10]  René Schoof,et al.  Hecke operators and the weight distributions of certain codes , 1991, Journal of Combinatorial Theory.

[11]  Aimo Tietäväinen,et al.  Codes and Number Theory , 1997 .

[12]  Nicholas M. Katz,et al.  A Note on Exponential Sums , 1995 .

[13]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[14]  Kalle Ranto,et al.  Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros , 2007, Finite Fields Their Appl..

[15]  G. Lachaud,et al.  The weights of the orthogonals of the extended quadratic binary Goppa codes , 1990, IEEE Trans. Inf. Theory.