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Register a new userA note on tight projective 2-designs
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Added by
Dustin Mixon
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for H^d determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of R^d(2d+1) of dimension 3.
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Consider two sequences of $n$ independent and identically distributed fair coin tosses, $X=(X_1,ldots,X_n)$ and $Y=(Y_1,ldots,Y_n)$, which are $rho$-correlated for each $j$, i.e. $mathbb{P}[X_j=Y_j] = {1+rhoover 2}$. We study the question of how large (small) the probability $mathbb{P}[X in A, Yin B]$ can be among all sets $A,Bsubset{0,1}^n$ of a given cardinality. For sets $|A|,|B| = Theta(2^n)$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|A|,|B| = 2^{Theta(n)}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $mathbb{P}[X in A, Yin B]$ in the regime of $rho to 1$. We also prove a similar tight lower bound, i.e. show that for $rhoto 0$ the pair of opposite Hamming balls approximately minimizes the probability $mathbb{P}[X in A, Yin B]$.
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs. Finally, we give the dimensions and lower bound of the minimum weights of these linear codes.
Projective Reed-Solomon (PRS) codes are Reed-Solomon codes of the maximum possible length q+1. The classification of deep holes --received words with maximum possible error distance-- for PRS codes is an important and difficult problem. In this paper, we use algebraic methods to explicitly construct three classes of deep holes for PRS codes. We show that these three classes completely classify all deep holes of PRS codes with redundancy at most four. Previously, the deep hole classification was only known for PRS codes with redundancy at most three in work arXiv:1612.05447
Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of narrow-sense BCH codes $mathcal{C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^m$ and established that the codewords of the minimum (or the second minimum) weight in these codes support infinite families of 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$-designs with $t=3,4$. In particular, we prove that the codewords of weight $7$ in $mathcal{C}_{(q,q+1,4,1)}$ support $4$-designs when $m geqslant 5$ is odd and $3$-designs when $m geqslant 4$ is even, which provide infinite classes of simple $t$-designs with new parameters. Another significant class of $t$-designs we produce in this paper has supplementary designs with parameters 4-$(2^{2s+1}+ 1,5,5)$; these designs have the smallest index among all the known simple 4-$(q+1,5,lambda)$ designs derived from codes for prime powers $q$; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL$(2,2^{2s+1})$ as automorphism group constructed by Alltop in 1969.
This note outlines the steps for proving that the moments of a randomly-selected subset of a general ETF (complex, with aspect ratio $0<gamma<1$) converge to the corresponding MANOVA moments. We bring here an extension for the proof of the Kesten-Mckay moments (real ETF, $gamma=1/2$) cite{magsino2020kesten}. In particular, we establish a recursive computation of the $r$th moment, for $r = 1,2,ldots$, and verify, using a symbolic program, that the recursion output coincides with the MANOVA moments.
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