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arXiv:2610.19459

Quantum LDPC Codes from Polynomial Minors

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

On Aqora:Vahid NouroziVahid Nourozi
quant-phmath.QASubmitted Oct 14, 2026• Announced Oct 15, 2026• 0 citations
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Abstract

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 $80/255$.

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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.
This paper gives a simple method to construct generator matrices with polynomial entries (and hence offers an alternative encoding method to the one commonly used) for all quasi-cyclic low-density parity-check (QC-LDPC) codes, even for those that are rank deficient. The approach is based on constructing a set of codewords with the desired total rank by using minors of the parity-check matrix. We exemplify the method on several well-known and standard codes. Moreover, we explore the connections between the minors of the parity-check matrix and the known upper bound on minimum distance and provide a method to compute the rank of any parity-check matrix representing a QC-LDPC code, and hence the dimension of the code, by using the minors of the corresponding polynomial parity-check matrix.
In this paper, we use 2-cyclotomic cosets of modulo n and generator polynomials to describe binary cyclic codes of length N=2αn with n odd. We discuss the conditions under which two cyclic codes [Formula: see text] and [Formula: see text] can be used to construct quantum codes by CSS construction or Steane's construction. Using the results of Chen, Promhouse and Tavares, and Castagnoli et al., we study the quantum codes that can be constructed from binary cyclic codes of length N=2αn with n odd and n≤99, and α≤2. We find that except the quantum codes constructed by Steane, there are also some very interesting quantum codes constructed from repeated-root cyclic codes, and some of the quantum codes constructed by Steane can be improved.
In this paper, we illustrate how to simplify the constructions of quantum error-correction codes via the quadratic residues. The suggested quantum code, which is the stabilizer quantum code generated from an Abelian group with the basis being commutative quantum operators selected from rows of an Pauli block matrix, does not require the dual-containing (or self-orthogonal) constraint necessary for the standard quantum error-correction code, thus allowing us to construct a quantum code with the large codeword length.
Quantum Bicycle LDPC Codes with High kd2/nkd^2/nkd2/n from Divisor-Driven Search
Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes [[42,12,4]]2[[42,12,4]]_2[[42,12,4]]2​ and [[62,12,4]]2[[62,12,4]]_2[[62,12,4]]2​ and produces a family of codes with competitive figure of merit kd2/nkd^2/nkd2/n, including [[66,20,7]]2[[66,20,7]]_2[[66,20,7]]2​ with kd2/n=14.85kd^2/n=14.85kd2/n=14.85, above the bivariate bicycle code [[144,12,12]]2[[144,12,12]]_2[[144,12,12]]2​ (kd2/n=12kd^2/n=12kd2/n=12) at less than half the block length, together with [[46,2,8]]2[[46,2,8]]_2[[46,2,8]]2​, [[66,2,9]]2[[66,2,9]]_2[[66,2,9]]2​, [[66,4,8]]2[[66,4,8]]_2[[66,4,8]]2​, [[66,6,8]]2[[66,6,8]]_2[[66,6,8]]2​ and, at n=90n=90n=90, [[90,16,6]]2[[90,16,6]]_2[[90,16,6]]2​, [[90,18,6]]2[[90,18,6]]_2[[90,18,6]]2​, [[90,20,6]]2[[90,20,6]]_2[[90,20,6]]2​. An exhaustive census at n=48n=48n=48 delineates the boundary of this picture: we exhibit a [[48,10,6]]2[[48,10,6]]_2[[48,10,6]]2​ code from a minimal 484848-element group (the Aydin--Tamo--Barg realization uses 727272 elements), and prove that distance 555 forces a stabilizer-rank loss, which excludes [[48,10,5]]2[[48,10,5]]_2[[48,10,5]]2​ from the weight-888 symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.
We propose two constructions, namely Construction 1 and Construction 2, for the design of protograph quantum LDPC codes by exploiting conventional quasi cyclic protograph LDPC codes with lift operation, and tensor product operation, respectively. We show that by carefully designing the proto-matrix using quadratic residue sets of parameters p = 4n ± 1, the proposed Construction 1 yields protograph quantum LDPC codes of various rates. Furthermore, by performing tensor product between two non-binary parity-check matrices obtained from idempotent polynomials of a quadratic residue set, the proposed Construction 2 yields a class of protograph quantum LDPC codes with rate at least 0.9. The performance of the proposed codes over quantum depolarizing channel with iterative sum-product decoding algorithm is shown.
Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of length at least six in their Tanner graphs. However, it is impossible to design a self-orthogonal parity check matrix of an LDPC code without introducing cycles of length four. In this paper, a new class of quantum LDPC codes based on lines and points of finite geometries is constructed. The parity check matrices of these codes are adapted to be self- orthogonal with containing only one cycle of length four in each pair of two rows. Also, the column and row weights, and bounds on the minimum distance of these codes are given. As a consequence, these codes can be encoded using shift-register encoding algorithms and can be decoded using iterative decoding algorithms over various quantum depolarizing channels.