Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal

• URL: http://arxiv.org/abs/2106.10180v3
• Date: Thu, 23 Dec 2021 15:49:47 GMT
• Title: Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal
• Abstract summary: We prove that there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n$ over $mathbb F_q2$. We also provide examples of Hermitian self-orthogonal $k$-dimensional Reed-Solomon codes of length $q2+1$ over $mathbb F_q2$, for $k=q-1$ and $q$ an odd power of two.
• Score: 0.7614628596146599
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove that there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n \leqslant q^2$ over ${\mathbb F}_{q^2}$ if and only if there is a polynomial $g \in {\mathbb F}_{q^2}$ of degree at most $(q-k)q-1$ such that $g+g^q$ has $q^2-n$ distinct zeros. This allows us to determine the smallest $n$ for which there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n$ over ${\mathbb F}_{q^2}$, verifying a conjecture of Grassl and R\"otteler. We also provide examples of Hermitian self-orthogonal $k$-dimensional generalised Reed-Solomon codes of length $q^2+1$ over ${\mathbb F}_{q^2}$, for $k=q-1$ and $q$ an odd power of two.

Related papers

• SVP Is NP-Hard for Some Rank-2 Cyclotomic Modules [17.893655677334575]
  We prove that the decision version of the Shortest Vector Problem ($mathrmSVP$) in the $ell binary$-norm is $mathrmNP$-complete on full-rank free sub-counts.<n>The construction also gives $mathrmNP$-hardness of search-$mathrmSVP$ under matrix-time Turing reductions.
  arXiv  Detail & Related papers  (2026-09-01T16:04:35Z)
• The Condition-Number Barrier in Sparse Least Squares [77.64108812086542]
  In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in convex optimization cannot be improved by a sparse-time algorithm.<n>We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis.
  arXiv  Detail & Related papers  (2026-08-03T17:57:01Z)
• Quantum Codes from $r$-Nearly Self-Orthogonal Linear Codes via Jordan Canonical Form over $\mathbb{F}_{q^2}$ [5.6342405886627]
  We introduce a Jordan-canonical-form framework for constructing $q$-ary quantum stabilizer codes from arbitrary classical linear codes over $F_q2$.<n>The framework does not require the classical linear code $mathcalC$ to satisfy the dual-containing condition (i.e., self-orthogonality)<n>Applying this construction to classical codes produces several record quantum codes that improve or supplement the best-known parameters in Grassl's tables.
  arXiv  Detail & Related papers  (2026-07-14T00:58:42Z)
• The Geometry Underlying the Quantum Harmonic Oscillator [51.56484100374058]
  We show that the eigenfunctions $_n$ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space $mathbb C2/mathbb Z_nsubsetmathbb C2$.<n>We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.
  arXiv  Detail & Related papers  (2026-04-23T07:38:35Z)
• Discrete symmetries in classical and quantum oscillators [51.56484100374058]
  We show the eigenfunctions $_n=zn$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation.<n>The superposition $=sum_n c_n_n$ arises only with incomplete knowledge of the initial data for solving the Schrdinger equation.
  arXiv  Detail & Related papers  (2026-01-05T10:04:39Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• More on genuine multi-entropy and holography [0.0]
  We give a prescription on how to construct $rm GM[mathttq]$ systematically for any $mathttq$. For general $mathttq$, $rm GM[mathttq]$ contains $N (mathttq)-p(mathttq-1)-1$ number of free parameters. These give $N (mathttq)+1$ number of new diagnostics for genuine $mathttq
  arXiv  Detail & Related papers  (2025-04-23T10:13:10Z)
• New estimates for character sums over sparse elements of finite fields [6.5990719141691825]
  We provide new estimates for the character sums $sum_ginmathcalGchi(f(g))$, where $mathcalG$ is a given sparse subsets of $mathbbF_qr$ and $f(X)$ is a sum over $mathbbF_qr$ of certain type.
  arXiv  Detail & Related papers  (2025-02-20T10:40:48Z)
• A Variant of the Bravyi-Terhal Bound for Arbitrary Boundary Conditions [10.560637835517094]
  We consider a quotient $mathbbZD/Lambda$ of $mathbbZD$ of cardinality $n$ on a $D$-dimensional lattice quotient. We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius $rho$, the minimum distance $d$ of the code satisfies $d leq msqrtgamma_D(sqrtD + 4rho)nfracD-1D$ whenever $n1/D
  arXiv  Detail & Related papers  (2025-02-07T15:18:40Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Some new infinite families of non-$p$-rational real quadratic fields [0.0]
  We give a simple methodology for constructing an infinite family of simultaneously non-$p_j$-rational real fields, unramified above any of the $p_j$. One feature of these techniques is that they may be used to yield fields $K=mathbbQ(sqrtD)$ for which a $p$-power cyclic component of the torsion group of the Galois groups of the maximal abelian pro-$p$-extension of $K$ unramified outside primes above $p$, is of size $pa
  arXiv  Detail & Related papers  (2024-06-20T18:00:51Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• Dimension-free Remez Inequalities and norm designs [48.5897526636987]
  A class of domains $X$ and test sets $Y$ -- termed emphnorm -- enjoy dimension-free Remez-type estimates. We show that the supremum of $f$ does not increase by more than $mathcalO(log K)2d$ when $f$ is extended to the polytorus.
  arXiv  Detail & Related papers  (2023-10-11T22:46:09Z)
• Higher rank antipodality [0.0]
  Motivated by general probability theory, we say that the set $X$ in $mathbbRd$ is emphantipodal of rank $k$. For $k=1$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee.
  arXiv  Detail & Related papers  (2023-07-31T17:15:46Z)
• Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates [0.0]
  We show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types are required to simulate it. The quantum circuit for implementing $mathcalU_cu(2otimes N)$ and $mathcalU_cd(Motimes N)$ are presented.
  arXiv  Detail & Related papers  (2022-09-11T06:24:24Z)
• The Approximate Degree of DNF and CNF Formulas [95.94432031144716]
  For every $delta>0,$ we construct CNF and formulas of size with approximate degree $Omega(n1-delta),$ essentially matching the trivial upper bound of $n. We show that for every $delta>0$, these models require $Omega(n1-delta)$, $Omega(n/4kk2)1-delta$, and $Omega(n/4kk2)1-delta$, respectively.
  arXiv  Detail & Related papers  (2022-09-04T10:01:39Z)
• Monogamy of entanglement between cones [68.8204255655161]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones. Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Mutually unbiased bases: polynomial optimization and symmetry [1.024113475677323]
  A set of $k$ orthonormal bases of $mathbb Cd$ is called mutually unbiased $|langle e,frangle |2 = 1/d$ whenever $e$ and $f$ are basis vectors in distinct bases. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them tractable.
  arXiv  Detail & Related papers  (2021-11-10T14:14:53Z)
• Tight Quantum Lower Bound for Approximate Counting with Quantum States [49.6558487240078]
  We prove tight lower bounds for the following variant of the counting problem considered by Aaronson, Kothari, Kretschmer, and Thaler ( 2020) The task is to distinguish whether an input set $xsubseteq [n]$ has size either $k$ or $k'=(1+varepsilon)k$.
  arXiv  Detail & Related papers  (2020-02-17T10:53:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.