In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting codes constructed in this paper are different from the ones in the literature.In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting codes constructed in this paper are different from the ones in the literature.

Wuhan University Journal of Natural SciencesFeb 1, 20240 citations

New Binary Quantum Codes Derived from Quasi-Twisted Codes with Hermitian Inner Product

Yu Yao, Yuena Ma, Jingjie Lv +2 more

In this paper, a special class of two-generator quasi-twisted (QT) codes with index 2 will be presented. We explore the algebraic structure of the class of QT codes and the form of their Hermitian dual codes. A sufficient condition for self-orthogonality with Hermitian inner product is derived. Using the class of Hermitian self-orthogonal QT codes, we construct two new binary quantum codes [[70, 42, 7]]2, [[78, 30, 10]]2. According to Theorem 6 of Ref.[2], we further can get 9 new binary quantum codes. So a total of 11 new binary quantum codes are obtained and there are 10 quantum codes that can break the quantum Gilbert-Varshamov (GV) bound.In this paper, a special class of two-generator quasi-twisted (QT) codes with index 2 will be presented. We explore the algebraic structure of the class of QT codes and the form of their Hermitian dual codes. A sufficient condition for self-orthogonality with Hermitian inner product is derived. Using the class of Hermitian self-orthogonal QT codes, we construct two new binary quantum codes [[70, 42, 7]]2, [[78, 30, 10]]2. According to Theorem 6 of Ref.[2], we further can get 9 new binary quantum codes. So a total of 11 new binary quantum codes are obtained and there are 10 quantum codes that can break the quantum Gilbert-Varshamov (GV) bound.

IEICE Transactions on Fundamentals of Electronics Communications and Computer SciencesMay 27, 20216 citations

The Metric Structure of Linear Codes

Diego Ruano

Springer eBooksJan 1, 20186 citations

Codes on finite geometries

Heng Tang, Jun Xu, Shu Lin +1 more

New algebraic methods for constructing codes based on hyperplanes of two different dimensions in finite geometries are presented. The new construction methods result in a class of multistep majority-logic decodable codes and three classes of low-density parity-check (LDPC) codes. Decoding methods for the class of majority-logic decodable codes, and a class of codes that perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs, are presented. Most of the codes constructed can be either put in cyclic or quasi-cyclic form and hence their encoding can be implemented with linear shift registers.New algebraic methods for constructing codes based on hyperplanes of two different dimensions in finite geometries are presented. The new construction methods result in a class of multistep majority-logic decodable codes and three classes of low-density parity-check (LDPC) codes. Decoding methods for the class of majority-logic decodable codes, and a class of codes that perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs, are presented. Most of the codes constructed can be either put in cyclic or quasi-cyclic form and hence their encoding can be implemented with linear shift registers.

IEEE Transactions on Information TheoryJan 24, 2005176 citations

Construction of a Family of Quantum Codes Using Sub-exceding Functions via the Hypergraph Product and the Generalized Shor Construction

Luc Rabefihavanana, Harinaivo Andriatahiny, Randriamiarampanahy Ferdinand

In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes $L_k$ and $L_k^{+}$ , defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters $[[6k^2,\, k^2,\, d]]$ . The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies $d=3$ for $k=3$ and $d=4$ for $k\ge4$ , establishing a new framework for structured quantum LDPC code design and optimization.In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes $L_k$ and $L_k^{+}$, defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters $[[6k^2,\, k^2,\, d]]$. The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies $d=3$ for $k=3$ and $d=4$ for $k\ge4$, establishing a new framework for structured quantum LDPC code design and optimization.

quant-pharXiv:2603.08213Mar 10, 20260 citations

In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes

L_k

and

L_k^{+}

, defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters

[[6k^2,\, k^2,\, d]]

. The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies

d=3

for

k=3

and

d=4

for

k\ge4

, establishing a new framework for structured quantum LDPC code design and optimization.