Some new results on dimension and Bose distance for various classes of BCH codes

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over ${\mathbb{F}}_q$ with designed distance $\delta =a{q}^{m-1}-1(\text{resp. }\delta =a\frac{q^m-1}{q-1})$ for all $1\le a\le q-1$, where q is a prime power and $m>1$ is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16], and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range $[a\frac{q^m-1}{q-1},a\frac{q^m-1}{q-1}+T]$ for $0\le a\le q-2$, where $T=q^{\frac{m+1}{2}}-1$ if m is odd, and $T=2q^{\frac{m}{2}}-1$ if m is even.     

New classes of optimal ternary cyclic codes with minimum distance four

Cyclic code is an interesting topic in coding theory and communication systems. In this paper, we investigate the ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$ based on some results proposed by Ding and Helleseth in 2013. Six new classes of optimal ternary cyclic codes are presented by determining the solutions of certain equations over ${\mathbb{F}}_{3^m}$.     

On the size of special class 1 graphs and (P3;k)-co-critical graphs

A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index ${\chi }^{\prime }(G)$ of G is Δ or $\mathrm{\Delta }+1$. A graph G is class 1 if ${\chi }^{\prime }(G)=\mathrm{\Delta }$, and class 2 if ${\chi }^{\prime }(G)=\mathrm{\Delta }+1$; G is Δ-critical if it is connected, class 2 and ${\chi }^{\prime }(G-e)<{\chi }^{\prime }(G)$ for every $e\in E(G)$. A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least $(n(\mathrm{\Delta }-1)+3)/2$ edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, ${\chi }^{\prime }(G+e)={\chi }^{\prime }(G)+1$ for every $e\in E(\bar{G})$. Such graphs have intimate relation to $(P_3;k)$-co-critical graphs, where a non-complete graph G is $(P_3;k)$-co-critical if there exists a k-coloring of $E(G)$ such that G does not contain a monochromatic copy of $P_3$ but every k-coloring of $E(G+e)$ contains a monochromatic copy of $P_3$ for every $e\in E(\bar{G})$. We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all $(P_3;k)$-co-critical graphs. We prove that if G is a $(P_3;k)$-co-critical graph on $n\ge k+2$ vertices, then$e(G)\ge \frac{k}{2}(n-\lceil \frac{k}{2}\rceil -\epsilon )+\left(\begin{array}{c}\lceil k/2\rceil +\epsilon \\ 2\end{array}\right),$ where ε is the remainder of $n-\lceil k/2\rceil$ when divided by 2. This bound is best possible for all $k\ge 1$ and $n\ge \lceil 3k/2\rceil +2$.     

Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.