72,105 research outputs found
Nearly optimal codebooks based on generalized Jacobi sums
Codebooks with small inner-product correlation are applied in many practical
applications including direct spread code division multiple access (CDMA)
communications, space-time codes and compressed sensing. It is extremely
difficult to construct codebooks achieving the Welch bound or the Levenshtein
bound. Constructing nearly optimal codebooks such that the ratio of its maximum
cross-correlation amplitude to the corresponding bound approaches 1 is also an
interesting research topic. In this paper, we firstly study a family of
interesting character sums called generalized Jacobi sums over finite fields.
Then we apply the generalized Jacobi sums and their related character sums to
obtain two infinite classes of nearly optimal codebooks with respect to the
Welch or Levenshtein bound. The codebooks can be viewed as generalizations of
some known ones and contain new ones with very flexible parameters
Linear codes with few weights over
In this paper, we construct an infinite family of five-weight codes from
trace codes over the ring , where The
trace codes have the algebraic structure of abelian codes. Their Lee weight is
computed by using character sums. Combined with Pless power moments and
Newton's Identities, the weight distribution of the Gray image of trace codes
was present. Their support structure is determined. An application to secret
sharing schemes is given.Comment: 14 pages, need help in page 12. arXiv admin note: text overlap with
arXiv:1612.0096
Index bounds for character sums with polynomials over finite fields
We provide an index bound for character sums of polynomials over finite
fields. This improves the Weil bound for high degree polynomials with small
indices, as well as polynomials with large indices that are generated by
cyclotomic mappings of small indices. As an application, we also give some
general bounds for numbers of solutions of some Artin-Schreier equations and
mininum weights of some cyclic codes
Deep Holes in Reed-Solomon Codes Based on Dickson Polynomials
For an Reed-Solomon code , it can be shown that any
received word lies a distance at most from , denoted
. Any word meeting the equality is called a deep
hole. Guruswami and Vardy (2005) showed that for a specific class of codes,
determining whether or not a word is a deep hole is NP-hard. They suggested
passingly that it may be easier when the evaluation set of is
large or structured. Following this idea, we study the case where the
evaluation set is the image of a Dickson polynomial, whose values appear with a
special uniformity. To find families of received words that are not deep holes,
we reduce to a subset sum problem (or equivalently, a Dickson
polynomial-variation of Waring's problem) and find solution conditions by
applying an argument using estimates on character sums indexed over the
evaluation set
Evaluation of the Hamming weights of a class of linear codes based on Gauss sums
Linear codes with a few weights have been widely investigated in recent
years. In this paper, we mainly use Gauss sums to represent the Hamming weights
of a class of -ary linear codes under some certain conditions, where is
a power of a prime. The lower bound of its minimum Hamming distance is
obtained. In some special cases, we evaluate the weight distributions of the
linear codes by semi-primitive Gauss sums and obtain some one-weight,
two-weight linear codes. It is quite interesting that we find new optimal codes
achieving some bounds on linear codes. The linear codes in this paper can be
used in secret sharing schemes, authentication codes and data storage systems
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
Two classes of linear codes with a few weights based on twisted Kloosterman sums
Linear codes with a few weights have wide applications in information
security, data storage systems, consuming electronics and communication
systems. Construction of the linear codes with a few weights and determination
of their parameters are an important research topic in coding theory. In this
paper, we construct two classes of linear codes with a few weights and
determine their complete weight enumerators based on twisted Kloosterman sums
Ternary codes associated with symplectic groups and power moments of Kloosterman sums with square arguments
In this paper, we construct two ternary linear codes associated with the
symplectic groups Sp(2,q) and Sp(4,q). Here q is a power of three. Then we
obtain recursive formulas for the power moments of Kloosterman sums with square
arguments and for the even power moments of those 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 symplectic groups
Sp(2n,q).Comment: No comment
Several classes of cyclic codes with either optimal three weights or a few weights
Cyclic codes with a few weights are very useful in the design of frequency
hopping sequences and the development of secret sharing schemes. In this paper,
we mainly use Gauss sums to represent the Hamming weights of a general
construction of cyclic codes. As applications, we obtain a class of optimal
three-weight codes achieving the Griesmer bound, which generalizes a Vega's
result in \cite{V1}, and several classes of cyclic codes with only a few
weights, which solve the open problem in \cite{V1}.Comment: 24 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
- β¦