Divisible Codes

Kurz, Sascha

doi:10.48550/arxiv.2112.11763

Cited by 4 publications

(5 citation statements)

References 75 publications

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“…In the theoretical parts we will also use classification for 2-divisible sets points that can e.g. be found in [13] or [22]. For the convenience of the reader we will also give a few selected proofs in Sect.…”

Section: Classification Of Tight Irreducible Avsps In Pg(n − 1 2) For...mentioning

confidence: 99%

Affine vector space partitions

Bamberg

Filmus

Ihringer

et al. 2023

Des. Codes Cryptogr.

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An affine vector space partition of $${{\,\textrm{AG}\,}}(n,q)$$ AG ( n , q ) is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.

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Section: Classification Of Tight Irreducible Avsps In Pg(n − 1 2) For...mentioning

confidence: 99%

Affine vector space partitions

Bamberg

Filmus

Ihringer

et al. 2023

Des. Codes Cryptogr.

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“…A linear code over a finite field with the Hamming metric is called ∆-divisible if the weights of all codewords are divisible by ∆. Surveys on divisible codes are given in [9,19]. Applications include subspace codes, partial spreads, vector space partitions, and distance optimal codes [19].…”

Section: Preliminariesmentioning

confidence: 99%

“…Surveys on divisible codes are given in [9,19]. Applications include subspace codes, partial spreads, vector space partitions, and distance optimal codes [19].…”

Section: Preliminariesmentioning

confidence: 99%

On Some Families of Codes Related to the Even Linear Codes Meeting the Grey–Rankin Bound

2022

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Bounds for the parameters of codes are very important in coding theory. The Grey–Rankin bound refers to the cardinality of a self-complementary binary code. Codes meeting this bound are associated with families of two-weight codes and other combinatorial structures. We study the relations among six infinite families of binary linear codes with two and three nonzero weights that are closely connected to the self-complementary linear codes meeting the Grey–Rankin bound. We give a construction method and partial classification results for such codes. The properties of the codes in the studied families and their relations help us in constructing codes of a higher dimension from codes with a given dimension.

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“…It is simple to manage congruence of weights in classical codes when the number of weights is small, and our approach creates an opportunity to design CSS codes by taking advantage of the extensive literature on classical codes with two or three weights (see [17]- [22]). In Section IV we construct new families of CSS codes using cosets of the first order Reed Muller code defined by quadratic forms.…”

Section: Introductionmentioning

confidence: 99%

Divisible Codes for Quantum Computation

Hu¹,

Liang²,

Calderbank³

2022

Preprint

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Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code C, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator UZ to preserve C and induce UL. The group of Z-stabilizers in a CSS code C is determined by the dual of a classical [n, k1] binary code C1, and the group of X-stabilizers is determined by a classical [n, k2] binary code C2 that is contained in C1. The requirement that a diagonal physical operator UZ fixes a CSS code C leads to constraints on the congruence of weights in cosets of C2. These constraints are a perfect fit to divisible codes, and represent an opportunity to take advantage of the extensive literature on classical codes with two or three weights. We construct new families of CSS codes using cosets of the first order Reed Muller code defined by quadratic forms. We provide a simple alternative to the standard method of deriving the coset weight distributions (based on Dickson normal form) that may be of independent interest. Finally, we develop an approach to circumventing the Eastin-Knill Theorem which states that no QECC can implement a universal set of logical gates through transversal gates alone. The essential idea is to design stabilizer codes in layers, with N1 inner qubits and N2 outer qubits, and to assemble a universal set of fault tolerant gates on the inner qubits.

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Divisible Codes

Cited by 4 publications

References 75 publications

Affine vector space partitions

Affine vector space partitions

On Some Families of Codes Related to the Even Linear Codes Meeting the Grey–Rankin Bound

Divisible Codes for Quantum Computation

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