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An $H$-decomposition of $G$ is a partition of the edge-set of $G$ into subsets, where each subset induces a copy of the graph $H$. A $k$-orthogonal $H$-decomposition of a graph $G$ is a set of $k$ $H$-decompositions of $G$, such that any two copies of $H$ in distinct $H$-decompositions intersect in at most one edge. When $G=K_v$ we call the $H$-decomposition an $H$-system of order $v$. In this paper we consider the case $H$ is an $l$-cycle and construct a pair of orthogonal $l$-cycle systems for all admissible orders when $l=5,6,7, 8\ or\ 9$, except $(l,v)=(7,7)$ and $(l,v)=(9,9)$.

We give a framework for generalizing LDPC code constructions that use Transversal Designs or related structures such as mutually orthogonal Latin squares. Our construction offers a broader range of code lengths and codes rates. Similar earlier constructions rely on the existence of finite fields of order a power of a prime. In contrast the LDPC codes constructed here are based on difference matrices and difference covering arrays, structures available for any order $a$. They satisfy the RC constraint and have, for $a$ odd, length $a^2$ and rate $1-\frac{4a-3}{a^2}$, and for $a$ even, length $a^2-a$ and rate at least $1-\frac{4a-6}{a^2-a}$. When $3$ does not divide $a$, these LDPC codes have stopping distance at least $8$. When $a$ is odd and both $3$ and $5$ do not divide $a$, our construction delivers an infinite family of QC-LDPC codes with minimum distance at least $10$. The simplicity of the construction allows us to theoretically verify these properties and analytically determine lower bounds for the minimum distance and stopping distance of the code. The BER and FER performance of our codes over AWGN (via simulation) is at the least equivalent to codes constructed previously, while in some cases significantly outperforming them.

In this paper we will show the existence of a face $2$-colourable biembedding of the complete graph onto an orientable surface where each face is a cycle of a fixed length $k$, for infinitely many values of $k$. In particular, under certain conditions, we show that there exists at least $(n-2)[(p-2)!]^2/(e^2 kn)$ non-isomorphic face $2$-colourable biembeddings of $K_{2nk+1}$ in which all faces are cycles of length $k=4p+3$. These conditions are: $n\equiv 1\mod 4$, $k\equiv 3\mod 4$ and either $n$ is prime or $n\gg k$ and $n\equiv 0\mod 3$ implies $p\equiv 1\mod 3$. To achieve this result we begin by verifying the existence of $(n-2)[(p-2)!/e]^2$ non-equivalent Heffter arrays, $H(n;k)$, which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo $2nk+1$; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arrays $H(n;k)$ that satisfy condition (1) was established earlier in \cite{BCDY} and in this current paper we vary this construction and show that there are at least $(n-2)[(p-2)!/e]^2$ such non-equivalent $H(n;k)$ that satisfy condition (1) and then show that each of these Heffter arrays also satisfy condition (2) under certain conditions., Comment: arXiv admin note: text overlap with arXiv:1906.07366

Square Heffter arrays are $n\times n$ arrays such that each row and each column contains $k$ filled cells, each row and column sum is divisible by $2nk+1$ and either $x$ or $-x$ appears in the array for each integer $1\leq x\leq nk$. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face $2$-colourable embedding of the complete graph $K_{2nk+1}$ on an orientable surface, where for each colour, the faces give a $k$-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo $2nk+1$; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) $k\equiv 0\mod 4$; or (b) $n\equiv 1\mod 4$ and $k\equiv 3\mod 4$; or (c) $n\equiv 0\mod 4$, $k\equiv3\mod 4$ and $n\gg k$. As corollaries to the above we obtain pairs of orthogonal $k$-cycle decompositions of $K_{2nk+1}$.

This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from difference covering arrays. While the original construction by Gallagher was by randomly allocating bits in a sparse parity-check matrix, over the past 20 years researchers have used a variety of more structured approaches to construct these codes, with the more recent constructions of well-structured LDPC coming from balanced incomplete block designs (BIBDs) and from Latin squares over finite fields. However these constructions have suffered from the limited orders for which these designs exist. Here we present a construction of LDPC codes of length $4n^2 - 2n$ for all $n$ using the cyclic group of order $2n$. These codes achieve high information rate (greater than 0.8) for $n \geq 8$, have girth at least 6 and have minimum distance 6 for $n$ odd., Comment: 11 pages

The strong form factor of the $B_{c} B_{c}J/\Psi$ vertex is calculated in the framework of the QCD sum rules method at finite temperature. Taking into account additional operators appearing at finite temperature, thermal Wilson expansion is obtained and QCD sum rules are derived. While increasing temperature, the strong form factor remains unchanged up to $T\simeq100~MeV$ but slightly increases after this point. After $T\simeq160~MeV$, the form factor suddenly decreases up to $T\simeq170~MeV$. The obtained result of the coupling constant by fitting the form factor at $Q^2=-m^2_{offshell}$ at $T=0$ is in a very good agreement with the QCD sum rules calculations at vacuum. Our prediction can be checked in the future experiments., Comment: 8 pages, 7 figures

In this paper we consider the uniformly resolvable decompositions of the complete graph $2K_v$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the cases in which all the resolution classes are either $C_4$ or $K_{1,3}$., Comment: arXiv admin note: text overlap with arXiv:1402.4603

In this study, we investigate the sensitivity of the masses and decay constants of the light $f_{2}(1270)$ and $a_{2}(1320)$ tensor mesons to the temperature using QCD sum rule approach. In our calculations, we take into account the additional operators appearing in operator product expansion at finite temperature. It is obtained that at deconfinement temperature the decay constants and masses decrease with amount of $6\%$ and $96\%$ compared to their vacuum values, respectively. Our results on the masses at zero temperature are consistent with the vacuum sum rules predictions as well as the experimental data., Comment: 4 Pages and 1 Table, Prepared for the proceeding of the 4th International Conference on Hadron Physics (TROIA'14)

The strong coupling constants among mesons are very important quantities as they can provide useful information on the nature of strong interaction among hadrons as well as the QCD vacuum. In this article, we investigate the strong vertices of the $D^{\ast}_{s}D_{s} \eta^{(\prime)}$ and $B^{\ast}_{s}B_{s} \eta^{(\prime)}$ in the framework of the QCD sum rule approach choosing the $\eta$ or $D_{s} (B_{s})$ meson as an off-shell state. We obtain the results $g_{D_s^*D_s\eta}=(1.46\pm 0.30)GeV^{-1}$, $g_{D_s^*D_s\eta^{\prime}}=(0.74\pm 0.16)GeV^{-1}$, $g_{B_s^*B_s\eta}=(5.29\pm 1.06)GeV^{-1}$ and $g_{B_s^*B_s\eta^{\prime}}=(2.29\pm 0.48)GeV^{-1}$ for the strong coupling constants under consideration, which can be checked in future experiments., Comment: 11 Pages, 4 Figures and 3 Tables

We show that a pair of orthogonal partial latin squares of order $n$ can be embedded in a pair of orthogonal latin squares of order at most $16n^4$ and all orders greater than or equal to $48n^4$. This paper provides the first direct polynomial order embedding construction in the literature.