173 research outputs found
On the weight distribution of second order Reed-Muller codes and their relatives
The weight distribution of second order -ary Reed-Muller codes have been
determined by Sloane and Berlekamp (IEEE Trans. Inform. Theory, vol. IT-16,
1970) for and by McEliece (JPL Space Programs Summary, vol. 3, 1969) for
general prime power . Unfortunately, there were some mistakes in the
computation of the latter one. This paper aims to provide a precise account for
the weight distribution of second order -ary Reed-Muller codes. In addition,
the weight distributions of second order -ary homogeneous Reed-Muller codes
and second order -ary projective Reed-Muller codes are also determined.Comment: 14 pages, Designs, Codes and Cryptography, Accepted, 201
The Minimum Distance of Some Narrow-Sense Primitive BCH Codes
Due to wide applications of BCH codes, the determination of their minimum
distance is of great interest. However, this is a very challenging problem for
which few theoretical results have been reported in the last four decades. Even
for the narrow-sense primitive BCH codes, which form the most well-studied
subclass of BCH codes, there are very few theoretical results on the minimum
distance. In this paper, we present new results on the minimum distance of
narrow-sense primitive BCH codes with special Bose distance. We prove that for
a prime power , the -ary narrow-sense primitive BCH code with length
and Bose distance , where , has minimum distance .
This is achieved by employing the beautiful theory of sets of quadratic forms,
symmetric bilinear forms and alternating bilinear forms over finite fields,
which can be best described using the framework of association schemes.Comment: 40 page
Constructions of Primitive Formally Dual Pairs Having Subsets with Unequal Sizes
The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann,
which reflects a remarkable symmetry among energy-minimizing periodic
configurations. This formal duality was later on translated into a purely
combinatorial property by Cohn, Kumar, Reiher and Sch\"urmann, where the
corresponding combinatorial objects were called formally dual pairs. Almost all
known examples of primitive formally dual pairs satisfy that the two subsets
have the same size. Indeed, prior to this work, there was only one known
example having subsets with unequal sizes in . Motivated by this example, we propose a lifting construction
framework and a recursive construction framework, which generate new primitive
formally dual pairs from known ones. As an application, for , we
obtain pairwise inequivalent primitive formally dual pairs in
, which have subsets with unequal sizes.Comment: Some corrections to version
The Weight Hierarchy of Some Reducible Cyclic Codes
The generalized Hamming weights (GHWs) of linear codes are fundamental
parameters, the knowledge of which is of great interest in many applications.
However, to determine the GHWs of linear codes is difficult in general. In this
paper, we study the GHWs for a family of reducible cyclic codes and obtain the
complete weight hierarchy in several cases. This is achieved by extending the
idea of \cite{YLFL} into higher dimension and by employing some interesting
combinatorial arguments. It shall be noted that these cyclic codes may have
arbitrary number of nonzeroes
On the Weight Distribution of Cyclic Codes with Niho Exponents
Recently, there has been intensive research on the weight distributions of
cyclic codes. In this paper, we compute the weight distributions of three
classes of cyclic codes with Niho exponents. More specifically, we obtain two
classes of binary three-weight and four-weight cyclic codes and a class of
nonbinary four-weight cyclic codes. The weight distributions follow from the
determination of value distributions of certain exponential sums. Several
examples are presented to show that some of our codes are optimal and some have
the best known parameters
Linking systems of difference sets
A linking system of difference sets is a collection of mutually related group
difference sets, whose advantageous properties have been used to extend
classical constructions of systems of linked symmetric designs. The central
problems are to determine which groups contain a linking system of difference
sets, and how large such a system can be. All previous constructive results for
linking systems of difference sets are restricted to 2-groups. We use an
elementary projection argument to show that neither the McFarland/Dillon nor
the Spence construction of difference sets can give rise to a linking system of
difference sets in non-2-groups. We make a connection to Kerdock and bent sets,
which provides large linking systems of difference sets in elementary abelian
2-groups. We give a new construction for linking systems of difference sets in
2-groups, taking advantage of a previously unrecognized connection with group
difference matrices. This construction simplifies and extends prior results,
producing larger linking systems than before in certain 2-groups, new linking
systems in other 2-groups for which no system was previously known, and the
first known examples in nonabelian groups.Comment: 25 pages. Better description of relationship to previous work,
especially in Section 3. Shorter proof of Proposition 1.7. Correction of
minor error
The Weight Distribution of a Class of Cyclic Codes Related to Hermitian Forms Graphs
The determination of weight distribution of cyclic codes involves evaluation
of Gauss sums and exponential sums. Despite of some cases where a neat
expression is available, the computation is generally rather complicated. In
this note, we determine the weight distribution of a class of reducible cyclic
codes whose dual codes may have arbitrarily many zeros. This goal is achieved
by building an unexpected connection between the corresponding exponential sums
and the spectrums of Hermitian forms graphs.Comment: 4 page
Narrow-Sense BCH Codes over \gf(q) with Length
Cyclic codes over finite fields are widely employed in communication systems,
storage devices and consumer electronics, as they have efficient encoding and
decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in
most cases among the best cyclic codes. A subclass of good BCH codes are the
narrow-sense BCH codes over \gf(q) with length . Little is
known about this class of BCH codes when . The objective of this paper is
to study some of the codes within this class. In particular, the dimension, the
minimum distance, and the weight distribution of some ternary BCH codes with
length are determined in this paper. A class of ternary BCH codes
meeting the Griesmer bound is identified. An application of some of the BCH
codes in secret sharing is also investigated
Some New Results on the Cross Correlation of -sequences
The determination of the cross correlation between an -sequence and its
decimated sequence has been a long-standing research problem. Considering a
ternary -sequence of period , we determine the cross correlation
distribution for decimations and , where .
Meanwhile, for a binary -sequence of period , we make an initial
investigation for the decimation , where is even and . It is shown that the cross correlation
takes at least four values. Furthermore, we confirm the validity of two famous
conjectures due to Sarwate et al. and Helleseth in this case
A Direct Construction of Primitive Formally Dual Pairs Having Subsets with Unequal Sizes
The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann,
which reflects a remarkable symmetry among energy-minimizing periodic
configurations. This formal duality was later translated into a purely
combinatorial property by Cohn, Kumar, Reiher and Sch\"urmann, where the
corresponding combinatorial objects were called formally dual pairs. So far,
except the results presented in Li and Pott (arXiv:1810.05433v3), we have
little information about primitive formally dual pairs having subsets with
unequal sizes. In this paper, we propose a direct construction of primitive
formally dual pairs having subsets with unequal sizes in , where . This construction recovers an infinite
family obtained in Li and Pott (arXiv:1810.05433v3), which was derived by
employing a recursive approach. Although the resulting infinite family was
known before, the idea of the direct construction is new and provides more
insights which were not known from the recursive approach.Comment: arXiv admin note: substantial text overlap with arXiv:1810.05433.
This version contains some minor corrections to version
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