Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.04617

New Methods for Constructing Classical and Quantum Codes from Graphs and Matroids

Anderson S. Barbosa, Franklin de L. Marquezino, Giuliano G. La Guardia

cs.DMquant-phSubmitted Oct 3, 2026• Announced Oct 6, 2026• 0 citations
arXivHTMLPDF

Abstract

In this paper, we present a method for constructing linear codes derived from book graphs. Applying such a method, we construct binary codes that have the greatest possible minimum distance for their length and dimension by means of the $2$-sum of two equal book graphs. We then utilize matroid and graph theory together to investigate some properties of the resulting linear codes. Furthermore, we show that, from the incidence matrix of any graph, we can obtain a linear code over any finite field maintaining the same parameters. Furthermore, we show how the condition for two linear codes to generate a Calderbank-Shor-Steane (CSS) code translates into matroid theory and, utilizing cycle graphs, we exhibit a way to generate codes that meet this condition.

Comments
19 pages, 4 figures

Related papers

Quantum codes and symplectic matroids

Pradeep Kiran Sarvepalli

The correspondence between linear codes and representable matroids is well known. But a similar correspondence between quantum codes and matroids is not known. We show that representable symplectic matroids over a finite field Fqcorrespond to Fq-linear quantum codes. This connection is straightforward but it does not appear to have been made earlier in literature. This correspondence is made through isotropic subspaces. We show that Calderbank-Shor-Steane (CSS) codes correspond to homogenous symplectic matroids while graph states, which figure so prominently in measurement based quantum computation, correspond to a special class of symplectic matroids, namely Lagrangian matroids. This association is useful in that it enables the study of symplectic matroids in terms of quantum codes and vice versa. Furthermore, it has application in the study of quantum secret sharing schemes.

arXiv:1104.1171•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2014•2 citations

Metodos de construção de codigos quanticos CSS e conexões entre codigos quanticos e matroides

Giuliano G. La Guardia

This thesis proposes, as the main contributions, constructions method of new families of quantum CSS codes.These constructions are based on classical cyclic codes of the types BCH, Reed-Solomon, Reed-Muller, Quadratic Residue and also are based on product codes of classical Reed-Solomon codes.The main family of quantum codes constructed in this work, i. e., quantum codes having better parameters, are the ones derived from classical BCH codes.Moreover, we present some new conditions in which quantum CSS cyclic codes are quantum MDS codes.In addition, we provide the elements to connect matroid theory and quantum coding theory.More specifically, we show that the weight enumerator of a CSS quantum code is equivalent to evaluating the Tutte polynomial of the direct sum of the matroid associated to the classical codes used in the CSS construction.

•Jul 12, 2008•0 citations

Good and asymptotically good quantum codes derived from algebraic geometry codes

Giuliano G. La Guardia, Francisco Revson F. Pereira

In this paper we construct several new families of quantum codes with good and asymptotically good parameters. These new quantum codes are derived from (classical) algebraic geometry (AG) codes by applying the Calderbank-Shor-Steane (CSS) construction. Many of these codes have large minimum distances when compared with its code length and they have relatively small Singleton defect. For example, we construct a family [[46; 2(t_2 - t1); d]]_25 of quantum codes, where t_1, t_2 are positive integers such that 1 < t_1 < t_2 < 23 and d >= min {46 - 2t_2, 2t_1-2}, of length n = 46, with minimum distance in the range 2 =< d =< 20, having Singleton defect four. Additionally, by utilizing t-point AG codes, with t >= 2, we show how to obtain sequences of asymptotically good quantum codes.

arXiv:1612.07150•Dec 21, 2016•2 citations

Quantum LDPC Codes from Polynomial Minors

Vahid Nourozi, Roxana Smarandache, David G. M. Mitchell

We construct quantum codes from codewords of a cyclic code and the minors of a polynomial matrix. The construction uses a two-row, three-column matrix whose minors yield a second matrix satisfying the CSS orthogonality condition. We derive the code dimension from the greatest common divisors of the entries and minors, with a simplified formula for odd lifting factors. A residue characterization identifies additional common factors of the minors and leads to a method for controlling the dimension through polynomial selection. We also give a condition for changing the minor generator without changing the code. Explicit polynomial constructions yield 13 quantum codes with verified minimum distances, including a $[[255,80,12]]$ code of rate $80/255$.

quant-pharXiv:2610.19459•Oct 15, 2026•0 citations

Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication

Yansheng Wu, Yoonjin Lee

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.

Mathematics•Sep 3, 2020•8 citations

On the Application of Error-Correcting Codes with Low-Density Generator Matrix over Different Quantum Channels

Hanqing Lou, Javier Garcia‐Frias

We propose the use of linear codes with low-density generator matrix in the context of error correction over different quantum channels. The basic idea is to construct a Calderbank-Shor-Steane (CSS) code based on the generator matrix and the parity-check matrix of an LDGM code, applying row operations in both matrices to achieve the desired quantum rate. Decoding is performed in an iterative manner, using message passing over the corresponding graphs. The proposed codes allow greater flexibility and are easier to design than existing sparse-graph quantum codes, while leading to better performance.

•Apr 3, 2006•23 citations
In this paper, we present a method for constructing linear codes derived from book graphs. Applying such a method, we construct binary codes that have the greatest possible minimum distance for their length and dimension by means of the 222-sum of two equal book graphs. We then utilize matroid and graph theory together to investigate some properties of the resulting linear codes. Furthermore, we show that, from the incidence matrix of any graph, we can obtain a linear code over any finite field maintaining the same parameters. Furthermore, we show how the condition for two linear codes to generate a Calderbank-Shor-Steane (CSS) code translates into matroid theory and, utilizing cycle graphs, we exhibit a way to generate codes that meet this condition.
The correspondence between linear codes and representable matroids is well known. But a similar correspondence between quantum codes and matroids is not known. We show that representable symplectic matroids over a finite field Fqcorrespond to Fq-linear quantum codes. This connection is straightforward but it does not appear to have been made earlier in literature. This correspondence is made through isotropic subspaces. We show that Calderbank-Shor-Steane (CSS) codes correspond to homogenous symplectic matroids while graph states, which figure so prominently in measurement based quantum computation, correspond to a special class of symplectic matroids, namely Lagrangian matroids. This association is useful in that it enables the study of symplectic matroids in terms of quantum codes and vice versa. Furthermore, it has application in the study of quantum secret sharing schemes.
This thesis proposes, as the main contributions, constructions method of new families of quantum CSS codes.These constructions are based on classical cyclic codes of the types BCH, Reed-Solomon, Reed-Muller, Quadratic Residue and also are based on product codes of classical Reed-Solomon codes.The main family of quantum codes constructed in this work, i. e., quantum codes having better parameters, are the ones derived from classical BCH codes.Moreover, we present some new conditions in which quantum CSS cyclic codes are quantum MDS codes.In addition, we provide the elements to connect matroid theory and quantum coding theory.More specifically, we show that the weight enumerator of a CSS quantum code is equivalent to evaluating the Tutte polynomial of the direct sum of the matroid associated to the classical codes used in the CSS construction.
In this paper we construct several new families of quantum codes with good and asymptotically good parameters. These new quantum codes are derived from (classical) algebraic geometry (AG) codes by applying the Calderbank-Shor-Steane (CSS) construction. Many of these codes have large minimum distances when compared with its code length and they have relatively small Singleton defect. For example, we construct a family [[46; 2(t_2 - t1); d]]_25 of quantum codes, where t_1, t_2 are positive integers such that 1 < t_1 < t_2 < 23 and d >= min {46 - 2t_2, 2t_1-2}, of length n = 46, with minimum distance in the range 2 =< d =< 20, having Singleton defect four. Additionally, by utilizing t-point AG codes, with t >= 2, we show how to obtain sequences of asymptotically good quantum codes.
We construct quantum codes from codewords of a cyclic code and the minors of a polynomial matrix. The construction uses a two-row, three-column matrix whose minors yield a second matrix satisfying the CSS orthogonality condition. We derive the code dimension from the greatest common divisors of the entries and minors, with a simplified formula for odd lifting factors. A residue characterization identifies additional common factors of the minors and leads to a method for controlling the dimension through polynomial selection. We also give a condition for changing the minor generator without changing the code. Explicit polynomial constructions yield 13 quantum codes with verified minimum distances, including a [[255,80,12]][[255,80,12]][[255,80,12]] code of rate 80/25580/25580/255.
It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.
We propose the use of linear codes with low-density generator matrix in the context of error correction over different quantum channels. The basic idea is to construct a Calderbank-Shor-Steane (CSS) code based on the generator matrix and the parity-check matrix of an LDGM code, applying row operations in both matrices to achieve the desired quantum rate. Decoding is performed in an iterative manner, using message passing over the corresponding graphs. The proposed codes allow greater flexibility and are easier to design than existing sparse-graph quantum codes, while leading to better performance.