On the resolutions of cyclic Steiner triple systems with small parameters

The paper presents useful invariants of resolutions of cyclic $STS(v)$ with $v\le 39$, namely of all resolutions of cyclic $STS(15)$, $STS(21)$ and $STS(27)$, of the resolutions with nontrivial automorphisms of cyclic $STS(33)$ and of resolutions with automorphisms of order $13$ of cyclic $STS(39)$

New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes

This paper presents several new construction techniques for low-density parity-check (LDPC) and systematic repeat-accumulate (RA) codes. Based on specific classes of combinatorial designs, the improved code design focuses on high-rate structured codes with constant column weights 3 and higher. The proposed codes are efficiently encodable and exhibit good structural properties. Experimental results on decoding performance with the sum-product algorithm show that the novel codes offer substantial practical application potential, for instance, in high-speed applications in magnetic recording and optical communications channels.Comment: 10 pages; to appear in "IEEE Transactions on Communications

Resolvable Mendelsohn designs and finite Frobenius groups

We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v, k, 1)$-Mendelsohn design for any integers $v > k \ge 2$ with $v \equiv 1 \mod k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\phi$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K \rtimes \langle \phi \rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v = p_{1}^{e_1} \ldots p_{t}^{e_t} \ge 3$ in prime factorization, and any prime $k$ dividing $p_{i}^{e_i} - 1$ for $1 \le i \le t$, there exists a resolvable perfect $(v, k, 1)$-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if $k$ is even and divides $p_{i} - 1$ for $1 \le i \le t$, then there are at least $\varphi(k)^t$ resolvable $(v, k, 1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\varphi$ is Euler's totient function.Comment: Final versio

Existence of APAV(q,k) with q a prime power ≡5(mod8) and k≡1(mod4)

AbstractStinson introduced authentication perpendicular arrays APAλ(t,k,v), as a special kind of perpendicular arrays, to construct authentication and secrecy codes. Ge and Zhu introduced APAV(q,k) to study APA1(2,k,v) for k=5, 7. Chen and Zhu determined the existence of APAV(q,k) with q a prime power ≡3(mod4) and odd k>1. In this article, we show that for any prime power q≡5(mod8) and any k≡1(mod4) there exists an APAV(q,k) whenever q>((E+E2+4F)/2)2, where E=[(7k−23)m+3]25m−3, F=m(2m+1)(k−3)25m and m=(k−1)/4

A visual representation of the Steiner triple systems of order 13

Steiner triple systems (STSs) are a basic topic in combinatorics. In an STS the elements can be collected in threes in such a way that any pair of elements is contained in a unique triple. The two smallest nontrivial STSs, with 7 and 9 elements, arise in the context of finite geometry and nonsingular cubic curves, and have well-known pictorial representations. On the contrary, an STS with 13 elements does not have an intrinsic geometric nature, nor a natural pictorial illustration. In this paper we present a visual representation of the two non-isomorphic Steiner triple systems of order 13 by means of a regular hexagram. The thirteen points of each system are the vertices of the twelve equilateral triangles inscribed in the hexagram. In the case of the non-cyclic system, our representation allows one to visualize in a simple, elegant and highly symmetric way the twenty-six triples, the six automorphisms and their orbits, the eight quadrilaterals, the ten mitres, the thirteen grids, the four 3-colouring patterns, the block-colouring and some distinguished ovals. Our construction is based on a very simple idea (seeing the blocks as much as possible as equilateral triangles), which can be further extended to get new representations of the STSs of order 7 and 9, and of one of the STSs of order 15

Equitable edge colored Steiner triple systems

A k-edge coloring of G is said to be equitable if the number of edges, at any vertex, colored with a certain color differ by at most one from the number of edges colored with a different color at the same vertex. An STS(v) is said to be polychromatic if the edges in each triple are colored with three different colors. In this paper, we show that every STS(v) admits a 3-edge coloring that is both polychromatic for the STS(v) and equitable for the underlying complete graph. Also, we show that, for v 1 or 3 (mod 6), there exists an equitable k-edge coloring of K which does not admit any polychromatic STS(v), for k = 3 and k = v - 2

Existence of r-fold perfect (v,K,1)-Mendelsohn designs with K⊆{4,5,6,7}

AbstractLet v be a positive integer and let K be a set of positive integers. A (v,K,1)-Mendelsohn design, which we denote briefly by (v,K,1)-MD, is a pair (X,B) where X is a v-set (of points) and B is a collection of cyclically ordered subsets of X (called blocks) with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t=1,2,…,r, every ordered pair of points of X are t-apart in exactly one block of B, then the (v,K,1)-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect (v,K,1)-MD. If K={k} and r=k−1, then an r-fold perfect (v,{k},1)-MD is essentially the more familiar (v,k,1)-perfect Mendelsohn design, which is briefly denoted by (v,k,1)-PMD. In this paper, we investigate the existence of r-fold perfect (v,K,1)-Mendelsohn designs for a specified set K which is a subset of {4, 5, 6, 7} containing precisely two elements