Related papers: The Spectrum of Quantum Caps in PG(4,4)
Let n > 1 be an integer, and let F denote a field of p elements for a prime p = 1 (mod n). By 2015, the question of existence or nonexistence of n-th power residue difference sets in F had been settled for all n < 24. We settle the case n =…
A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate…
In this survey we concern ourself with the question, wether there exists a fix-free code for a given sequence of codeword lengths. For a given alphabet, we obtain the {\em Kraftsum} of a code, if we divide for every length the number of…
This paper demonstrates that complex PT-symmetric periodic potentials possess real band spectra. However, there are significant qualitative differences in the band structure for these potentials when compared with conventional real periodic…
The $k$-spectrum is the set of all $\alpha>0$ such that $G(n,n^{-\alpha})$ does not obey the 0-1 law for FO sentences with quantifier depth at most $k$. In this paper, we prove that the minimum $k$ such that the $k$-spectrum is infinite…
We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals $(2 \sqrt{2},3)$ and $[-3,-2)$ achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on…
A $4$-general set in ${\rm PG}(n,q)$ is a set of points of ${\rm PG}(n,q)$ spanning the whole ${\rm PG}(n,q)$ and such that no four of them are on a plane. Such a pointset is said to be complete if it is not contained in a larger…
We prove that the interval $(5/6, 1)$ contains no 3-dimensional canonical thresholds.
In this paper, some nonbinary quantum codes using classical codes over Gaussian integers are obtained. Also, some of our quantum codes are better than or comparable with those known before, (for instance [[8; 2; 5]]4+i).
An (n,1,p)-Quantum Random Access (QRA) coding, introduced by Ambainis, Nayak, Ta-shma and Vazirani in ACM Symp. on Theory of Computing 1999, is the following communication system: The sender which has n-bit information encodes his/her…
In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD$(n,2)$ over $\mathbb{F}_3$ and $\mathbb{F}_4$. We…
Recently, it was shown that the spectrum of anomalous dimensions and other important observables in N = 4 SYM are encoded into a simple nonlinear Riemann-Hilbert problem: the P\mu-system or Quantum Spectral Curve. In this letter we present…
In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in $\mathbb{Z}^{n}$ with radius $r\ge2$ and dimension $n\ge3$. Besides its own interest in coding theory and discrete geometry, this conjecture is also…
A system of linear constraints can be unsatisfiable and yet admit a solution in the form of quantum observables whose correlated outcomes satisfy the constraints. Recently, it has been claimed that such a satisfiability gap can be…
We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by…
We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the…
All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact…
Consider a quantum particle trapped between a curved layer of constant width built over a complete, non-compact, $\mathcal C^2$ smooth surface embedded in $\mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense that…
In Ref. [1] the authors claim that the Almost Quantum set of correlations cannot reproduce two points on the boundary of the quantum set of correlations. This claim is incorrect. The underlying issue is that the associated SDP is not…
Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely…