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Air Force Engineering University

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Selected work

Representative Papers

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

Aug 10, 2026

This work addresses the challenge of efficiently determining the dimension and distance of quantum Bicycle LDPC codes, which traditionally rely on group-algebraic constructions that hinder the discovery of high-performance short codes. The authors reformulate the construction of cyclic Bicycle codes as a purely algebraic problem in the polynomial ring $\mathbb{F}_2[x]/(x^\ell - 1)$. Exploiting the fact that self-orthogonality is automatically satisfied, they show that the code dimension is directly given by a polynomial greatest common divisor, while the distance can be precisely verified via the Calderbank correspondence mapping to additive codes over $\mathbb{F}_4$. This enables efficient pre-screening and systematic enumeration, bypassing group-theoretic limitations. For the first time, an exhaustive algebraic search of Bicycle codes becomes feasible, revealing the onset boundary of the co-set phenomenon and yielding new high-performance codes such as $[[66,20,7]]_2$ ($kd^2/n=14.85$) and $[[46,2,8]]_2$. A complete census at $n=48$ disproves the existence of a $[[48,10,5]]_2$ code, thereby clarifying fundamental construction limits.

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Quantum Codes from $r$-Nearly Self-Orthogonal Linear Codes via Jordan Canonical Form over $\mathbb{F}_{q^2}$

Jul 13, 2026

This work addresses the limitation in traditional constructions of quantum stabilizer codes, which require classical codes to satisfy Hermitian self-orthogonality. To overcome this constraint, the authors introduce the novel notion of “$r$-nearly self-orthogonal” codes. Starting from an arbitrary classical linear code, they explicitly construct self-orthogonal codes by leveraging Jordan canonical form decomposition, analysis of Hermitian dual spaces, and rank-one perturbation techniques. They further establish a sufficient criterion that guarantees preservation of the minimum distance. The resulting $q$-ary quantum codes achieve parameters $[[n+r, 2k-n+r, \geq d]]_q$, with several concrete instances either surpassing or complementing the best-known codes listed in Grassl’s table.

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Latest Papers

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

Aug 10, 2026

This work addresses the challenge of efficiently determining the dimension and distance of quantum Bicycle LDPC codes, which traditionally rely on group-algebraic constructions that hinder the discovery of high-performance short codes. The authors reformulate the construction of cyclic Bicycle codes as a purely algebraic problem in the polynomial ring $\mathbb{F}_2[x]/(x^\ell - 1)$. Exploiting the fact that self-orthogonality is automatically satisfied, they show that the code dimension is directly given by a polynomial greatest common divisor, while the distance can be precisely verified via the Calderbank correspondence mapping to additive codes over $\mathbb{F}_4$. This enables efficient pre-screening and systematic enumeration, bypassing group-theoretic limitations. For the first time, an exhaustive algebraic search of Bicycle codes becomes feasible, revealing the onset boundary of the co-set phenomenon and yielding new high-performance codes such as $[[66,20,7]]_2$ ($kd^2/n=14.85$) and $[[46,2,8]]_2$. A complete census at $n=48$ disproves the existence of a $[[48,10,5]]_2$ code, thereby clarifying fundamental construction limits.

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Quantum Codes from $r$-Nearly Self-Orthogonal Linear Codes via Jordan Canonical Form over $\mathbb{F}_{q^2}$

Jul 13, 2026

This work addresses the limitation in traditional constructions of quantum stabilizer codes, which require classical codes to satisfy Hermitian self-orthogonality. To overcome this constraint, the authors introduce the novel notion of “$r$-nearly self-orthogonal” codes. Starting from an arbitrary classical linear code, they explicitly construct self-orthogonal codes by leveraging Jordan canonical form decomposition, analysis of Hermitian dual spaces, and rank-one perturbation techniques. They further establish a sufficient criterion that guarantees preservation of the minimum distance. The resulting $q$-ary quantum codes achieve parameters $[[n+r, 2k-n+r, \geq d]]_q$, with several concrete instances either surpassing or complementing the best-known codes listed in Grassl’s table.

0 citationsRead paper