20 research outputs found
Weight Distributions of Hamming Codes
We derive a recursive formula determing the weight distribution of the
[n=(q^m-1)/(q-1), n-m, 3] Hamming code H(m,q), when (m, q-1)=1. Here q is a
prime power. The proof is based on Moisio's idea of using Pless power moment
identity together with exponential sum techniques
Recursive formulas generating power moments of multi-dimensional Kloosterman sums and -multiple power moments of Kloosterman sums
In this paper, we construct two binary linear codes associated with
multi-dimensional and multiple power Kloosterman sums (for any fixed )
over the finite field . Here 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 -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.Comment: 14 page
Weight Distributions of Hamming Codes (II)
In a previous paper, we derived a recursive formula determining the weight
distributions of the [n=(q^m-1)/(q-1)] Hamming code H(m,q), when (m,q-1)=1.
Here q is a prime power. We note here that the formula actually holds for any
positive integer m and any prime power q, without the restriction (m, q-1)=1
Simple Recursive Formulas Generating Power Moments of Kloosterman Sums
In this paper, we construct four binary linear codes closely connected with
certain exponential sums over the finite field F_q and F_q-{0,1}. Here q is a
power of two. Then we obtain four recursive formulas for the power moments of
Kloosterman sums in terms of the frequencies of weights in the codes. This is
done via Pless power moment identity and by utilizing the explicit expressions
of the exponential sums obtained earlier.Comment: 8 page
A Recursive Formula for Power Moments of 2-Dimensional Kloosterman Sums Assiciated with General Linear Groups
In this paper, we construct a binary linear code connected with the
Kloosterman sum for . Here is a power of two. Then we obtain a
recursive formula generating the power moments 2-dimensional Kloosterman sum,
equivalently that generating the even power moments of Kloosterman sum in terms
of the frequencies of weights in the code. This is done via Pless power moment
identity and by utilizing the explicit expression of the Kloosterman sum for
.Comment: 9 page
Codes Associated with and Power Moments of Kloosterman Sums with Trace One Arguments
We construct a binary linear code , associated with the orthogonal
group . Here is a power of two. Then we obtain a recursive formula
for the odd power moments of Kloosterman sums with trace one arguments in terms
of the frequencies of weights in the codes and . This
is done via Pless power moment identity and by utilizing the explicit
expressions of Gauss sums for the orthogonal groups
Codes Associated with Orthogonal Groups and Power Moments of Kloosterman Sums
In this paper, we construct three binary linear codes ,
, , respectively associated with the orthogonal
groups , , , with powers of two. Then
we obtain recursive formulas for the power moments of Kloosterman and
2-dimensional Kloosterman sums in terms of the frequencies of weights in the
codes. This is done via Pless power moment identity and by utilizing the
explicit expressions of Gauss sums for the orthogonal groups. We emphasize
that, when the recursive formulas for the power moments of Kloosterman sums are
compared, the present one is computationally more effective than the previous
one constructed from the special linear group . We illustrate our
results with some examples
Codes Associated with Special Linear Groups and Power Moments of Multi-dimensional Kloosterman Sums
In this paper, we construct the binary linear codes associated
with finite special linear groups , with both \emph{n,q} powers of
two. Then, via Pless power moment identity and utilizing our previous result on
the explicit expression of the Gauss sum for , we obtain a recursive
formula for the power moments of multi-dimensional Kloosterman sums in terms of
the frequencies of weights in . In particular, when , this
gives a recursive formula for the power moments of Kloosterman sums. We
illustrate our results with some examples
On Binary Cyclic Codes with Five Nonzero Weights
Let , , be odd and .
In this paper the value distribution of following exponential sums
\sum\limits_{x\in \bF_q}(-1)^{\mathrm{Tr}_1^n(\alpha x^{2^{2k}+1}+\beta
x^{2^k+1}+\ga x)}\quad(\alpha,\beta,\ga\in \bF_{q}) is determined. As an
application, the weight distribution of the binary cyclic code \cC, with
parity-check polynomial where , and
are the minimal polynomials of , and
respectively for a primitive element of \bF_q, is
also determined
Gold type codes of higher relative dimension
Some new Gold type codes of higher relative dimension are introduced. Their
weight distribution is determined