Recursive formulas generating power moments of multi-dimensional Kloosterman sums and \(m\)-multiple power moments of Kloosterman sums

Dae San Kim

Abstract


In this paper, we construct two binary linear codes associated with
multi-dimensional and \(m\)-multiple power Kloosterman sums (for any
fixed \(m\)) over the finite field \(\mathbb{F}_{q}\). Here \(q\) is a
power of two. The former codes are dual to a subcode of the binary
hyper-Kloosterman code. Then we obtain two recursive formulas for
the power moments of multi-dimensional Kloosterman sums and for the
\(m\)-multiple power moments of Kloosterman sums in terms of the
frequencies of weights in the respective codes. This is done via
Pless power moment identity and yields, in the case of power moments
of multi-dimensional Kloosterman sums, much simpler recursive
formulas than those associated with finite special linear groups
obtained previously.

Keywords


Index terms-recursive formula, multi-dimensional Kloosterman sum,

Full Text:

PDF

References


L. Carlitz, Gauss sums over finite fields of order 2n, Acta. Arith. 15(1969), 247–265.

L. Carlitz, A note on exponential sums, Pacific J. Math. 30(1969), 35–37.

R. J. Evans, Seventh power moments of Kloosterman sums, Israel J. Math.

(2010), 349–362.

K. Hulek, J. Spandaw, B. van Geemen, and D. van Straten, The modularity of

the Barth-Nieto quintic and its relatives, Adv. Geom. 1(2001), 263-289.

D. S. Kim, Codes associated with special linear groups and power moments of

multi-dimensional Kloosterman sums, Ann. Mat. Pura Appli. 190(2011), 61-76.

D. S. Kim, Infinite families of recursive formulas generating power moments of

ternary Kloosterman sums with square arguments arising from symplectic groups,

Adv. Math. Commun. 3(2009), 167–178.

D. S. Kim, Codes associated with O+(2n, 2r) and power moments of Kloosterman

sums, Integers 12(2012), 237–257.

H. D. Kloosterman, On the representation of numbers in the form ax2 + by2 +

cz2 + dt2, Acta Math. 49(1926), 407-464.

G. Lachaud and J. Wolfmann, The weights of the orthogonals of the extended

quadratic binary Goppa codes, IEEE Trans. Inform. Theory 36(1990), 686-692.

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia Math. Appl.20, Cambridge

University Pless, Cambridge, 1987.

R. Livn´e, Motivic orthogonal two-dimensional representations of Gal(Q/Q), Israel

J. Math. 92(1995), 149-156.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes,

North-Holland, Amsterdam, 1998.

M. Moisio, The moments of a Kloosterman sum and the weight distribution of a

Zetterberg-type binary cyclic code, IEEE Trans. Inform. Theory 53(2007), 843-847.

M. Moisio, On the duals of binary hyper-Kloosterman codes, SIAM J. Disc. Math.

(2008), 273-287.

M. Moisio, K. Ranto, M. Rinta-aho and V¨a¨an¨anen, On the weight distribution of the

duals of irreducible cyclic codes, cyclic codes with two zeros and hyper-Kloosterman

codes, Adv. Appl. Discrete Math. 3(2009), 155-164.

C. Peters, J. Top, and M. van der Vlugt, The Hasse zeta function of a K3 surface

related to the number of words of weight 5 in the Melas codes, J. Reine Angew.

Math. 432(1992), 151-176.

H. Sali´e, ¨U ber die Kloostermanschen Summen S(u, v; q),Math. Z. 34(1931), 91-109.

R. Schoof and M. van der Vlugt, Hecke operators and the weight distributions of

certain codes, J. Combin. Theory Ser. A 57(1991), 163-186.


Refbacks

  • There are currently no refbacks.