Distance-Regular Graphs

A classical method of constructing a linear code over $ {\mathrm {GF}}(q)$ with a $t$ -design is to use the incidence matrix of the $t$ -design as a generator matrix over $ {\mathrm {GF}}(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented.

Linear Codes From Some 2-Designs

A classical method of constructing a linear code over $\gf(q)$ with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over $\gf(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented.

Linear Codes from Some 2-Designs

Weight distributions of cyclic codes with respect to pairwise coprime order elements

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, let $\Bbb F_r$ be a finite field with $r=q^m$ . Suppose that $g_1, g_2 \in \Bbb F_r^*$ are not conjugates over $\Bbb F_q$ , $ \operatorname {ord}(g_1)=n_1$ , $ \operatorname {ord}(g_2)=n_2$ , $d=\gcd (n_1, n_2)$ , and $n={n_1n_2}/ d$ . Let $\Bbb F_q(g_1)=\Bbb F_{q^{m_1}}, \Bbb F_q(g_2)=\Bbb F_{q^{m_2}}$ , and $\text{T}_i$ denote the trace function from $\Bbb F_{q^{m_i}}$ to $\Bbb F_q$ for $i=1, 2$ . We define a cyclic code $\mathcal C_{(q, m, n_1, n_2)}=\{c(a,b) : a \in \Bbb F_{q^{m_1}}, b \in \Bbb F_{q^{m_2}}\},$ where $c(a,b)=(\text {T}_1(ag_1^0)+\text {T}_2(bg_2^0), \text {T}_1(ag_1^1)+\text {T}_2(bg_2^1), \ldots , \text {T}_1(ag_1^{n-1})+\text {T}_2(bg_2^{n-1})).$ We mainly use Gauss periods to present the weight distribution of the cyclic code $\mathcal C_{(q, m, n_1, n_2)}$ . As applications, we determine the weight distribution of cyclic code $\mathcal C_{(q, m,q^{m_1}-1,q^{m_2}-1)}$ with $\gcd (m_1,m_2)=1$ ; in particular, it is a three-weight cyclic code if $\gcd (q-1,m_1-m_2)=1$ . We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes.

Hamming Weights of the Duals of Cyclic Codes With Two Zeros

Cameron-Liebler line classes with parameter x = q 2 - 1 2

A conflict-avoiding code (CAC) $C$ of length $n$ with weight $k$ is a family of binary sequences of length $n$ and weight $k$ satisfying $\sum_{0\le t\le n-1}x_{it}x_{j,t+s}\le \lambda$ for any distinct codewords $x_i=(x_{i0},x_{i1},\ldots,x_{i,n-1})$ and $x_j=(x_{j0},x_{j1},\ldots,x_{j,n-1})$ in $C$ and for any integer $s$, where the subscripts are taken modulo $n$. A CAC with maximum code size for given $n$ and $k$ is said to be optimal. A CAC has been studied for sending messages correctly through a multiple-access channel. The use of an optimal CAC enables the largest possible number of potential users to transmit information efficiently and reliably. In this paper, the case $\lambda=1$ is treated, and various direct and recursive constructions of optimal CACs for weight $k=4$ and $5$ are obtained by providing constructions of CACs for general weight $k$. In particular, the maximum code size of CACs satisfying certain sufficient conditions is determined through number theoretical and combinatorial approaches.

Constant Weight Conflict-Avoiding Codes

In (M. Buratti, J Combin Des 7:406---425, 1999), Buratti pointed out the lack of systematic treatments of constructions for relative difference families. The concept of strong difference families was introduced to cover such a problem. However, unfortunately, only a few papers consciously using the useful concept have appeared in the literature in the past 10 years. In this paper, strong difference families, difference covers and their connections with relative difference families and optical orthogonal codes are discussed.

Strong difference families, difference covers, and their applications for relative difference families

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of $${\mathbb{Z}}_n$$ such that $$\Delta(x) \cap \Delta(y) = \emptyset$$ holds for any $$x,y\in C$$ , $$x\not= y$$ , where $$\Delta(x)=\{j-i\,|\, i,j\in x, i\not= j\}$$ . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if $$\bigcup_{x\in C}\Delta(x)={\mathbb{Z}}_n\setminus \{0\}$$ holds and any codeword $$x\in C$$ has the form $$\{0,i,2i,\ldots,(k-1)i\}$$ . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n's admitting the condition through a number theoretical approach.

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes of finite fields, and they generalize the constructions given by Feng and Xiang [10,12]. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.

Koji Momihara

Papers

Cameron-Liebler line classes with parameter x = q 2 - 1 2

Constant Weight Conflict-Avoiding Codes

Strong difference families, difference covers, and their applications for relative difference families

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes