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Register a new userOptimal optical orthogonal signature pattern codes with weight three and cross-correlation constraint one
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Optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention as signature patterns of spatial optical code division multiple access networks. In this paper, an improved upper bound on the size of an $(m,n,3,lambda_a,1)$-OOSPC with $lambda_a=2,3$ is established. The exact number of codewords of an optimal $(m,n,3,lambda_a,1)$-OOSPC is determined for any positive integers $m,nequiv2 ({rm mod } 4)$ and $lambda_ain{2,3}$.
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Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access (CDMA) network for 2-dimensional image transmission. There is a one-to-one correspondence between an $(m, n, w, lambda)$-OOSPC and a $(lambda+1)$-$(mn,w,1)$ packing design admitting an automorphism group isomorphic to $mathbb{Z}_mtimes mathbb{Z}_n$. In 2010, Sawa gave the first infinite class of $(m, n, 4, 2)$-OOSPCs by using $S$-cyclic Steiner quadruple systems. In this paper, we use various combinatorial designs such as strictly $mathbb{Z}_mtimes mathbb{Z}_n$-invariant $s$-fan designs, strictly $mathbb{Z}_mtimes mathbb{Z}_n$-invariant $G$-designs and rotational Steiner quadruple systems to present some constructions for $(m, n, 4, 2)$-OOSPCs. As a consequence, our new constructions yield more infinite families of optimal $(m, n, 4, 2)$-OOSPCs. Especially, we shall see that in some cases an optimal $(m, n, 4, 2)$-OOSPC can not achieve the Johnson bound.
In this paper we investigate combinatorial constructions for $w$-cyclic holely group divisible packings with block size three (briefly by $3$-HGDPs). For any positive integers $u,v,w$ with $uequiv0,1~(bmod~3)$, the exact number of base blocks of a maximum $w$-cyclic $3$-HGDP of type $(u,w^v)$ is determined. This result is used to determine the exact number of codewords in a maximum three-dimensional $(utimes vtimes w,3,1)$ optical orthogonal code with at most one optical pulse per spatial plane and per wavelength plane.
How can $d+k$ vectors in $mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $theta(d,k):=min_Xmax_{x eq yin X}|langle x,yrangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $Xsubseteqmathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $dtoinfty$. In establishing bounds on $theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1choose 2}$ equiangular lines in $mathbb{R}^k$. Using this connection, we are able to pin down $theta(d,k)$ whenever $kin{1,2,3,7,23}$ and establish asymptotics for general $k$. The main tool is an upper bound on $mathbb{E}_{x,ysimmu}|langle x,yrangle|$ whenever $mu$ is an isotropic probability mass on $mathbb{R}^k$, which may be of independent interest. Our results translate naturally to the analogous question in $mathbb{C}^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $mathbb{C}^k$, also known as SIC-POVM in physics literature.
The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singletons defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.
We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary $k$-dimensional linear code $mathcal{C}$ ($k = 2,3,4$) into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d ge d(mathcal{C})$. For $k > 4$, we suggest a recursive method to embed a $k$-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length $n$ with dimension 4 and any length $n otequiv 6,13,14,21,22,28,29 pmod{31}$ with dimension 5. We determine the exact optimal minimum distances of $[n,4]$ self-orthogonal codes which were left open by Li-Xu-Zhao (2008) when $n equiv 0,3,4,5,10,11,12 pmod{15}$. Then, using MAGMA, we observe that our embedding sends an optimal linear code to an optimal self-orthogonal code.
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