Hamming Weights in Irreducible Cyclic Codes

Irreducible cyclic codes are an interesting type of codes and have applications in space communications. They have been studied for decades and a lot of progress has been made. The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes

Binary linear codes with at most 4 weights

For the past decades, linear codes with few weights have been widely studied, since they have applications in space communications, data storage and cryptography. In this paper, a class of binary linear codes is constructed and their weight distribution is determined. Results show that they are at most 4-weight linear codes. Additionally, these codes can be used in secret sharing schemes.Comment: 8 page

On an open problem about a class of optimal ternary cyclic codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth. Let $C_{(1,e)}$ be a cyclic code of length $3^m-1$ over GF(3) with two nonzeros $\alpha$ and $\alpha^e$, where $\alpha$ is a generator of $GF(3^m)^*$ and e is a given integer. It is shown that $C_{(1,e)}$ is optimal with parameters $[3^m-1,3^m-1-2m,4]$ if one of the following conditions is met. 1) $m\equiv0(\mathrm{mod}~ 4)$, $m\geq 4$, and $e=3^\frac{m}{2}+5$. 2) $m\equiv2(\mathrm{mod}~ 4)$, $m\geq 6$, and $e=3^\frac{m+2}{2}+5$

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

A class of optimal ternary cyclic codes and their duals

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let $m=2\ell+1$ for an integer $\ell\geq 1$ and $\pi$ be a generator of \gf(3^m)^*. In this paper, a class of cyclic codes \C_{(u,v)} over \gf(3) with two nonzeros $\pi^{u}$ and $\pi^{v}$ is studied, where $u=(3^m+1)/2$, and $v=2\cdot 3^{\ell}+1$ is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over \gf(3^m), the cyclic code \C_{(u,v)} is shown to have minimal distance four, which is the best minimal distance for any linear code over \gf(3) with length $3^m-1$ and dimension $3^m-1-2m$ according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied

A Class of Six-weight Cyclic Codes and Their Weight Distribution

In this paper, a family of six-weight cyclic codes over GF(p) whose duals have two zeros is presented, where p is an odd prime. And the weight distribution of these cyclic codes is determined.Comment: arXiv admin note: text overlap with arXiv:1302.0952, arXiv:1302.0569, arXiv:1301.4824 by other author

Gold type codes of higher relative dimension

Some new Gold type codes of higher relative dimension are introduced. Their weight distribution is determined

A note on the five valued conjectures of Johansen and Helleseth and zeta functions

For the complete five-valued cross-correlation distribution between two $m$-sequences ${s_t}$ and ${s_{dt}}$ of period $2^m-1$ that differ by the decimation $d={{2^{2k}+1}\over {2^k+1}}$ where $m$ is odd and \mbox{gcd}(k,m)=1, Johansen and Hellseth expressed it in terms of some exponential sums. And two conjectures are presented that are of interest in their own right. In this correspondence we study these conjectures for the particular case where $k=3$, and the cases $k=1,2$ can also be analyzed in a similar process. When $k>3$, the degrees of the relevant polynomials will become higher. Here the multiplicity of the biggest absolute value of the cross-correlation is no more than one-sixth of the multiplicity corresponding the smallest absolute value.Comment: 14 page

Linear code derived from the primage of quadratic function

Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power $q$, we construct some class of linear code over finite field $\mathbb{F}_q$ with defining set be the preimage of general quadratic form function and determine the explicit complete weight enumerators of the linear codes. Our construction cover all the corresponding result with quadratic form function and they may have applications in cryptography and secret sharing schemes.Comment: arXiv admin note: text overlap with arXiv:1506.06830 by other author

A Class of Linear Codes With Three Weights

Linear codes have been an interesting subject of study for many years. Recently, linear codes with few weights have been constructed and extensively studied. In this paper, for an odd prime p, a class of three-weight linear codes over Fp are constructed. The weight distributions of the linear codes are settled. These codes have applications in authentication codes, association schemes and data storage systems.Comment: 11 pages,2 table