Multi-twisted (MT) codes were introduced as a generalization of quasi-twisted (QT) codes. QT codes have been known to contain many good codes. In this work, we show that codes with good parameters and desirable properties can be obtained from MT codes. These include best known and optimal classical codes with additional properties such as reversibility and self-duality, and new and best known non-binary quantum codes obtained from special cases MT codes. Often times best known quantum codes in the literature are obtained indirectly by considering extension rings. Our constructions have the advantage that we obtain these codes by more direct and simpler methods. Additionally, we found theoretical results about binomials over finite fields that are useful in our search.Multi-twisted (MT) codes were introduced as a generalization of quasi-twisted (QT) codes. QT codes have been known to contain many good codes. In this work, we show that codes with good parameters and desirable properties can be obtained from MT codes. These include best known and optimal classical codes with additional properties such as reversibility and self-duality, and new and best known non-binary quantum codes obtained from special cases MT codes. Often times best known quantum codes in the literature are obtained indirectly by considering extension rings. Our constructions have the advantage that we obtain these codes by more direct and simpler methods. Additionally, we found theoretical results about binomials over finite fields that are useful in our search.

arXiv:2008.07037Aug 17, 20201 citation

(2,2)-GB Codes: Classification and Comparison With Weight-4 Surface Codes

François Arnault, Philippe Gaborit, Nicolas Saussay

Generalized Bicycle (GB) codes offer a compelling alternative to surface codes for quantum error correction. This paper focuses on (2,2)-Generalized Bicycle codes, constructed from pairs of binary circulant matrices with two non-zero elements per row. Leveraging a lower bound on their minimum distance, we construct three novel infinite families of optimal (2,2)-GB codes with parameters [[2n2, 2,n]], [[4r2, 2, 2r]], and [[(2t+1)2+1, 2, 2t+1]]. These families match the performance of Kitaev’s toric code and the best 2D weight-4 surface codes, reaching known theoretical limits. In particular, the second family breaks a long-held belief by providing optimal even-distance GB codes, previously deemed impossible. All are CSS codes derived from Cayley graphs. Recognizing that standard equivalence relations do not preserve their CSS structure, we introduce a CSS-preserving equivalence relation for rigorous comparison of Cayley graph-based CSS codes. Under this framework, the first two families are inequivalent to all previously known optimal weight-4 2D surface codes, while the third family is equivalent to the best-known odd-distance 2D surface code. Finally, we classify all extremal, non-equivalent (2, 2)-GB codes with length below 200 and present a comparison table with existing notable 2D weight-4 surface codes.Generalized Bicycle (GB) codes offer a compelling alternative to surface codes for quantum error correction. This paper focuses on (2,2)-Generalized Bicycle codes, constructed from pairs of binary circulant matrices with two non-zero elements per row. Leveraging a lower bound on their minimum distance, we construct three novel infinite families of optimal (2,2)-GB codes with parameters [[2n2, 2,n]], [[4r2, 2, 2r]], and [[(2t+1)2+1, 2, 2t+1]]. These families match the performance of Kitaev’s toric code and the best 2D weight-4 surface codes, reaching known theoretical limits. In particular, the second family breaks a long-held belief by providing optimal even-distance GB codes, previously deemed impossible. All are CSS codes derived from Cayley graphs. Recognizing that standard equivalence relations do not preserve their CSS structure, we introduce a CSS-preserving equivalence relation for rigorous comparison of Cayley graph-based CSS codes. Under this framework, the first two families are inequivalent to all previously known optimal weight-4 2D surface codes, while the third family is equivalent to the best-known odd-distance 2D surface code. Finally, we classify all extremal, non-equivalent (2, 2)-GB codes with length below 200 and present a comparison table with existing notable 2D weight-4 surface codes.

arXiv:2507.21237IEEE Transactions on Information TheoryMay 7, 20260 citations

Optimal binary codes and binary construction of quantum codes

Weiliang, Wang, Yangyu +3 more

Frontiers of Computer ScienceJan 1, 20141 citation

Quantum codes with bounded minimum distance

Carlos Galindo, Fernando Hernando, Diego Ruano

Jan 1, 20152 citations

Engineering Quantum Error Correction Codes Using Evolutionary Algorithms

Mark A. Webster, Dan E. Browne

Quantum error correction and the use of quantum error correction codes are likely to be essential for the realization of practical quantum computing. Because the error models of quantum devices vary widely, quantum codes that are tailored for a particular error model may have much better performance. In this work, we present a novel evolutionary algorithm that searches for an optimal stabilizer code for a given error model, number of physical qubits, and number of encoded qubits. We demonstrate an efficient representation of stabilizer codes as binary strings, which allows for random generation of valid stabilizer codes as well as mutation and crossing of codes. Our algorithm finds stabilizer codes whose distance closely matches the best-known-distance codes of Grassl (2007) for $n \leq 20$ physical qubits. We perform a search for optimal distance Calderbank–Steane–Shor codes and compare their distance to the best known codes. Finally, we show that the algorithm can be used to optimize stabilizer codes for biased error models, demonstrating a significant improvement in the undetectable error rate for $[[12,1]]_{2}$ codes versus the best-known-distance code with the same parameters. As part of this work, we also introduce an evolutionary algorithm QDistEvol for finding the distance of quantum error correction codes.Quantum error correction and the use of quantum error correction codes are likely to be essential for the realization of practical quantum computing. Because the error models of quantum devices vary widely, quantum codes that are tailored for a particular error model may have much better performance. In this work, we present a novel evolutionary algorithm that searches for an optimal stabilizer code for a given error model, number of physical qubits, and number of encoded qubits. We demonstrate an efficient representation of stabilizer codes as binary strings, which allows for random generation of valid stabilizer codes as well as mutation and crossing of codes. Our algorithm finds stabilizer codes whose distance closely matches the best-known-distance codes of Grassl (2007) for$n \leq 20$physical qubits. We perform a search for optimal distance Calderbank–Steane–Shor codes and compare their distance to the best known codes. Finally, we show that the algorithm can be used to optimize stabilizer codes for biased error models, demonstrating a significant improvement in the undetectable error rate for$[[12,1]]_{2}$codes versus the best-known-distance code with the same parameters. As part of this work, we also introduce an evolutionary algorithm QDistEvol for finding the distance of quantum error correction codes.

arXiv:2409.13017IEEE Transactions on Quantum EngineeringJan 1, 20256 citations

n \leq 20

physical qubits. We perform a search for optimal distance Calderbank–Steane–Shor codes and compare their distance to the best known codes. Finally, we show that the algorithm can be used to optimize stabilizer codes for biased error models, demonstrating a significant improvement in the undetectable error rate for

[[12,1]]_{2}

codes versus the best-known-distance code with the same parameters. As part of this work, we also introduce an evolutionary algorithm QDistEvol for finding the distance of quantum error correction codes.