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تسجيل مستخدم جديدA note on explicit constructions of designs
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An $(n,r,s)$-system is an $r$-uniform hypergraph on $n$ vertices such that every pair of edges has an intersection of size less than $s$. Using probabilistic arguments, R{o}dl and v{S}iv{n}ajov{a} showed that for all fixed integers $r> s ge 2$, there exists an $(n,r,s)$-system with independence number $Oleft(n^{1-delta+o(1)}right)$ for some optimal constant $delta >0$ only related to $r$ and $s$. We show that for certain pairs $(r,s)$ with $sle r/2$ there exists an explicit construction of an $(n,r,s)$-system with independence number $Oleft(n^{1-epsilon}right)$, where $epsilon > 0$ is an absolute constant only related to $r$ and $s$. Previously this was known only for $s>r/2$ by results of Chattopadhyay and Goodman
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Unitary $t$-designs are `good finite subsets of the unitary group $U(d)$ that approximate the whole unitary group $U(d)$ well. Unitary $t$-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of
quantum information science. If a unitary $t$-design itself is a group then it is called a unitary $t$-group. Although it is known that unitary $t$-designs in $U(d)$ exist for any $t$ and $d$, the unitary $t$-groups do not exist for $tgeq 4$ if $dgeq 3$, as it is shown by Guralnick-Tiep (2005) and Bannai-Navarro-Rizo-Tiep (BNRT, 2018). Explicit constructions of exact unitary $t$-designs in $U(d)$ are not easy in general. In particular, explicit constructions of unitary $4$-designs in $U(4)$ have been an open problem in quantum information theory. We prove that some exact unitary $(t+1)$-designs in the unitary group $U(d)$ are constructed from unitary $t$-groups in $U(d)$ that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary $3$-designs in $U(3)$ from the unitary $2$-group $SL(3,2)$ in $U(3),$ and also unitary $4$-designs in $U(4)$ from the unitary $3$-group $Sp(4,3)$ in $U(4)$ numerically. We also discuss some related problems.
The purpose of this paper is to give explicit constructions of unitary $t$-designs in the unitary group $U(d)$ for all $t$ and $d$. It seems that the explicit constructions were so far known only for very special cases. Here explicit construction mea
ns that the entries of the unitary matrices are given by the values of elementary functions at the root of some given polynomials. We will discuss what are the best such unitary $4$-designs in $U(4)$ obtained by these methods. Indeed we give an inductive construction of designs on compact groups by using Gelfand pairs $(G,K)$. Note that $(U(n),U(m) times U(n-m))$ is a Gelfand pair. By using the zonal spherical functions for $(G,K)$, we can construct designs on $G$ from designs on $K$. We remark that our proofs use the representation theory of compact groups crucially. We also remark that this method can be applied to the orthogonal groups $O(d)$, and thus provides another explicit construction of spherical $t$-designs on the $d$ dimensional sphere $S^{d-1}$ by the induction on $d$.
Recently, Cohen, Haeupler and Schulman gave an explicit construction of binary tree codes over polylogarithmic-sized output alphabet based on Pudl{a}ks construction of maximum-distance-separable (MDS) tree codes using totally-non-singular triangular
matrices. In this short note, we give a unified and simpler presentation of Pudl{a}k and Cohen-Haeupler-Schulmans constructions.
We study tight projective 2-designs in three different settings. In the complex setting, Zauners conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach
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In the picture-hanging puzzle we are to hang a picture so that the string loops around $n$ nails and the removal of any nail results in a fall of the picture. We show that the length of a sequence representing an element in the free group with $n$ ge
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