Abstract:
Supporting Information for the Paper: Optimal Ternary Constant-Composition Codes of Weight Four and Distance Five, IEEE Trans. Inform. Theory, To Appear.

Abstract:
Let $\mathcal{X}$ be an $n$-element set. Assume $\mathscr{F}$ is a collection of subsets of $\mathcal{X}$. We call $\mathscr{F}$ an $r$-cover-free family if $F_0\nsubseteq F_1\cup\cdots\cup F_r$ holds for all distinct $F_0,F_1,...,F_r\in\mathscr{F}$. Given $r$, denote $n(r)$ the minimal $n$ such that there exits an $r$-cover-free family on an $n$-element set with cardinality larger than $n$. Thirty years ago, Erd\H{o}s, Frankl and F{\"u}redi \cite{CFF} proved that $\binom{r+2}{2}\leq n(r)< r^2+o(r^2)$. They also conjectured $\lim_{r\rightarrow\infty} n(r)/r^2=1$ and claimed that $n(r)>(1+o(1))\frac{5}{6}r^2$, without proof. In this paper, it is proved that $\lim_{r\rightarrow\infty} n(r)/r^2\geq(15+\sqrt{33})/24$, which is a quantity in $[6/7,7/8]$. In particular, their conjecture is proved to be true for all r-cover-free families with uniform $(r+1)$-subsets.

Abstract:
The sizes of optimal constant-composition codes of weight three have been determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes played an important role in their constructions. In this paper, we study the problem of constructing optimal quaternary constant-composition codes with Hamming weight four and minimum distances five or six through group divisible codes and Room square approaches. The problem is solved leaving only five lengths undetermined. Previously, the results on the sizes of such quaternary constant-composition codes were scarce.

Abstract:
For a Gray code in the scheme of rank modulation for flash memories, the codewords are permutations and two consecutive codewords are obtained using a push-to-the-top operation. We consider snake-in-the-box codes under Kendall's $\tau$-metric, which is a Gray code capable of detecting one Kendall's $\tau$-error. We answer two open problems posed by Horovitz and Etzion. Firstly, we prove the validity of a construction given by them, resulting in a snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+1$. Secondly, we come up with a different construction aiming at a longer snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+3$. The construction is applied successfully to $S_7$.

Abstract:
One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. In this paper, we first present a construction of classical codes based on certain class of polynomials. Through these classical codes, we are able to obtain some new quantum codes. It turns out that some of quantum codes exhibited here have better parameters than the ones available in the literature. Meanwhile, we give a new class of quantum synchronizable codes with highest possible tolerance against misalignment from duadic codes.

Abstract:
One of the central tasks in quantum error-correction is to construct quantum codes that have good parameters. In this paper, we construct three new classes of quantum MDS codes from classical Hermitian self-orthogonal generalized Reed-Solomon codes. We also present some classes of quantum codes from matrix-product codes. It turns out that many of our quantum codes are new in the sense that the parameters of quantum codes cannot be obtained from all previous constructions.

Abstract:
In this paper, we construct an infinite series of 9-class association schemes from a refinement of the partition of Delsarte-Goethals codes by their Lee weights. The explicit expressions of the dual schemes are determined through direct manipulations of complicated exponential sums. As a byproduct, the other three infinite families of association schemes are also obtained as fusion schemes and quotient schemes.

Abstract:
Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. In this paper, we give two new constructions of quantum MDS convolutional codes derived from generalized Reed-Solomon codes and obtain eighteen new classes of quantum MDS convolutional codes. Most of them are new in the sense that the parameters of the codes are different from all the previously known ones.

Abstract:
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.

Abstract:
The locally repairable codes (LRCs) were introduced to correct erasures efficiently in distributed storage systems. LRCs are extensively studied recently. In this paper, we first deal with the open case remained in \cite{q} and derive an improved upper bound for the minimum distances of LRCs. We also give an explicit construction for LRCs attaining this bound. Secondly, we consider the constructions of LRCs with any locality and availability which have high code rate and minimum distance as large as possible. We give a graphical model for LRCs. By using the deep results from graph theory, we construct a family of LRCs with any locality $r$ and availability $2$ with code rate $\frac{r-1}{r+1}$ and optimal minimum distance $O(\log n)$ where $n$ is the length of the code.