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arXiv:2606.08771

Algebra of Bivariate-Bicycle Surface Codes

Renyu Wang, Leonid P. Pryadko

quant-phcs.ITSubmitted Jun 7, 2026• Announced Jun 9, 2026• 0 citations
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Abstract

We relate the properties of bivariate-bicycle-surface (BBS) codes, constructed from a pair of bivariate polynomials over a finite field, to the number and location of their common roots in the extension field. The number of roots $(x,y)$ with finite, non-zero coordinates -- counted with algebraic multiplicity -- determines the dimension of the codes. This dimension is invariant under monomial automorphisms of the Laurent polynomial ring. Conversely, roots with zero or infinite $x$- or $y$-coordinates indicate that specialized generators are required near the corresponding boundary (e.g., the left or right boundary for a root where $x$ is zero or infinite, respectively). These roots can appear or disappear under monomial transformations, which reveals the structure of tilted boundaries. Based on these results, we formulate a prescription for constructing BBS codes that works for regions with rectangular, diagonal, and arbitrarily tilted boundaries. A key advantage of this approach is that no corner corrections are needed, provided the polynomials satisfy orientation-specific edge conditions.

Comments
22 pages, 8 figures included

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Bivariate Bicycle Codes over Group Algebras

Chaobin Liu

We study a weighted-shift formulation of bivariate bicycle group-algebra (BBGA) codes over finite group algebras. One binary shift acts by right multiplication and the other by left multiplication; associativity ensures that these shifts commute even for a nonabelian seed group. Bivariate polynomials in the shifts, including mixed monomials, therefore define CSS codes. We give explicit sparsity bounds and distinguish nonabelian seed groups from codes that are intrinsically inequivalent to abelian realizations. For $D_3$, a pair of weighted permutation shifts generates a regular $C_6\times C_{12}$ action and recovers the known $[[144,12,12]]$ and $[[144,14,14]]$ bivariate bicycle codes. We also present a computer-verified $[[144,16,12]]$ code over $A_4\times C_6$, with every check of weight eight. This example belongs to the established nonabelian two-block group-algebra family. Exhaustive low-weight enumeration and an intrinsic stabilizer-support invariant certify that it is inequivalent to any ordinary abelian BB or two-block group-algebra realization, even under qubit permutations combined with local Clifford operations. Finally, a nonuniform weighted-shift construction over $\mathbb{F}_2[C_3]$ yields a $[[18,4,3]]$ code with check weights six and eight. An exact enumeration of its minimum stabilizer supports rules out every ordinary binary 2BGA realization at that length, under the same equivalence. This establishes that the unrestricted weighted-shift formulation extends beyond the ordinary 2BGA class, even with an abelian seed group.

quant-pharXiv:2609.36213•Sep 30, 2026•0 citations

Spectral Theory of Semisimple Bivariate Bicycle Codes

Eric Sabo, Mahir Bilen Can, David Marquis

Extending the classical theory of two-dimensional cyclic codes, we develop an algebraic approach to bivariate bicycle codes. Using Frobenius-orbit idempotents, formulas for logical dimensions are derived and lower bounds on minimum distances are established. A systematic theory of code symmetries is formulated to construct a structured block-monomial subgroup of coordinate permutations. Several explicit examples show how to generate these codes from first principles without relying on numerical searches. An appendix extends the analysis to BCH-based product constructions.

quant-pharXiv:2608.27565•Aug 31, 2026

Sequences of Bivariate Bicycle Codes from Covering Graphs

Benjamin C. B. Symons, Abhishek Rajput, Dan E. Browne

We show that given an instance of a bivariate bicycle (BB) code, it is possible to generate an infinite sequence of new BB codes using increasingly large covering graphs of the original code's Tanner graph. When a BB code has a Tanner graph that is a $h$-fold covering of the base BB code's Tanner graph, we refer to it as a $h$-\textit{cover code}. We show that for a BB code to be a $h$-cover code, its lattice parameters and defining polynomials must satisfy simple algebraic conditions relative to those of the base code. By extending the graph covering map to a chain map, we show there are induced projection and lifting maps on (co)homology that enable the projection and lifting of logical operators and, in certain cases, automorphisms between the base and the cover code. The search space of cover codes is considerably reduced compared to the full space of possible polynomials and we find that many interesting examples of BB codes, such as the $[[144,12,12]]$ gross code, can be viewed as cover codes. We also apply our method to search for BB codes with weight 8 checks and find many new codes, including a $[[64,14,8]]$ and $[[144,14,14]]$ code. For an $h$-cover code of an $[[n,k,d]]$ BB code with parameters $[[n_h = hn, k_h, d_h]]$, we prove that $k_h \geq k$ for any $h$, and $d_h \leq hd$ when $h$ is odd. Furthermore if $h$ is odd and $k_h = k$, we prove the distance lower bound $d \leq d_h$. We conjecture it is always true that an $h$-cover BB code of a base $[[n,k,d]]$ BB code obeys the distance bounds $d \leq d_h \leq hd$. While the focus of this work is on bivariate bicycle codes, we expect these methods to generalise readily to many group algebra codes and to certain code constructions involving hypergraph, lifted, and balanced products.

quant-pharXiv:2511.13560•Nov 18, 2025•0 citations

Qudit Twisted-Torus Codes in the Bivariate Bicycle Framework

Mourad Halla

We study finite-length qudit quantum low-density parity-check (LDPC) codes from translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. Recent qubit work [PRX Quantum 6, 020357 (2025)] showed that, within the bivariate-bicycle viewpoint, twisting generalized toric patterns can significantly improve the finite-size distance of qubit codes. Building on this insight, we extend the construction and the search to qudit codes over finite fields. Using algebraic methods based on Laurent-polynomial ideals and Gröbner-basis computations over $\mathbb{F}_q$, we compute the number of logical qudits on any twisted torus and identify compact codes with favorable rate--distance tradeoffs. Overall, for the finite sizes explored, twisted-torus qudit constructions typically achieve larger estimated distances than their untwisted counterparts and, at the same number of physical and logical qudits, larger estimated distances than previously reported twisted qubit instances. The best new codes are tabulated.

quant-pharXiv:2602.04443•Feb 5, 2026•0 citations

Independent Trivariate Bicycle Codes

Aygul Azatovna Galimova

We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a $[[140,6,14]]$ code with $kd^2/n = 8.40$. In the code-capacity setting, the $[[140,6,14]]$ code achieves a pseudothreshold of $8.0\%$ and $kd^2/n = 8.40$, exceeding the best multivariate bicycle code of Voss et al.\ ($7.9\%$, $kd^2/n = 2.67$). With circuit-level depolarizing noise, pseudothresholds reach $0.59\%$ for $[[140,6,14]]$ and $0.53\%$ for $[[84,6,10]]$. On the SI1000 superconducting noise model, the $[[140,6,14]]$ code achieves a per-round per-observable rate of $5.6 \times 10^{-5}$ at $p = 0.20\%$. We additionally present two self-dual codes with weight-8 stabilizers: $[[54,14,5]]$ ($kd^2/n = 6.48$) and $[[128,20,8]]$ ($kd^2/n = 10.0$). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.

quant-pharXiv:2603.17703•Mar 19, 2026•0 citations

Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes

Mohammad Rowshan, Simon Devitt

Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$-logical quotient $K/S$ over $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$. Using the Frobenius structure of the finite group algebra, we prove $r_{\mathcal A}=r_{\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\mathcal C}=10$ and $d_{\mathcal A}=12$.

quant-pharXiv:2609.39395•Oct 1, 2026•0 citations
We relate the properties of bivariate-bicycle-surface (BBS) codes, constructed from a pair of bivariate polynomials over a finite field, to the number and location of their common roots in the extension field. The number of roots (x,y)(x,y)(x,y) with finite, non-zero coordinates -- counted with algebraic multiplicity -- determines the dimension of the codes. This dimension is invariant under monomial automorphisms of the Laurent polynomial ring. Conversely, roots with zero or infinite xxx- or yyy-coordinates indicate that specialized generators are required near the corresponding boundary (e.g., the left or right boundary for a root where xxx is zero or infinite, respectively). These roots can appear or disappear under monomial transformations, which reveals the structure of tilted boundaries. Based on these results, we formulate a prescription for constructing BBS codes that works for regions with rectangular, diagonal, and arbitrarily tilted boundaries. A key advantage of this approach is that no corner corrections are needed, provided the polynomials satisfy orientation-specific edge conditions.
We study a weighted-shift formulation of bivariate bicycle group-algebra (BBGA) codes over finite group algebras. One binary shift acts by right multiplication and the other by left multiplication; associativity ensures that these shifts commute even for a nonabelian seed group. Bivariate polynomials in the shifts, including mixed monomials, therefore define CSS codes. We give explicit sparsity bounds and distinguish nonabelian seed groups from codes that are intrinsically inequivalent to abelian realizations. For D3D_3D3​, a pair of weighted permutation shifts generates a regular C6×C12C_6\times C_{12}C6​×C12​ action and recovers the known [[144,12,12]][[144,12,12]][[144,12,12]] and [[144,14,14]][[144,14,14]][[144,14,14]] bivariate bicycle codes. We also present a computer-verified [[144,16,12]][[144,16,12]][[144,16,12]] code over A4×C6A_4\times C_6A4​×C6​, with every check of weight eight. This example belongs to the established nonabelian two-block group-algebra family. Exhaustive low-weight enumeration and an intrinsic stabilizer-support invariant certify that it is inequivalent to any ordinary abelian BB or two-block group-algebra realization, even under qubit permutations combined with local Clifford operations. Finally, a nonuniform weighted-shift construction over F2[C3]\mathbb{F}_2[C_3]F2​[C3​] yields a [[18,4,3]][[18,4,3]][[18,4,3]] code with check weights six and eight. An exact enumeration of its minimum stabilizer supports rules out every ordinary binary 2BGA realization at that length, under the same equivalence. This establishes that the unrestricted weighted-shift formulation extends beyond the ordinary 2BGA class, even with an abelian seed group.
Extending the classical theory of two-dimensional cyclic codes, we develop an algebraic approach to bivariate bicycle codes. Using Frobenius-orbit idempotents, formulas for logical dimensions are derived and lower bounds on minimum distances are established. A systematic theory of code symmetries is formulated to construct a structured block-monomial subgroup of coordinate permutations. Several explicit examples show how to generate these codes from first principles without relying on numerical searches. An appendix extends the analysis to BCH-based product constructions.
We show that given an instance of a bivariate bicycle (BB) code, it is possible to generate an infinite sequence of new BB codes using increasingly large covering graphs of the original code's Tanner graph. When a BB code has a Tanner graph that is a hhh-fold covering of the base BB code's Tanner graph, we refer to it as a hhh-\textit{cover code}. We show that for a BB code to be a hhh-cover code, its lattice parameters and defining polynomials must satisfy simple algebraic conditions relative to those of the base code. By extending the graph covering map to a chain map, we show there are induced projection and lifting maps on (co)homology that enable the projection and lifting of logical operators and, in certain cases, automorphisms between the base and the cover code. The search space of cover codes is considerably reduced compared to the full space of possible polynomials and we find that many interesting examples of BB codes, such as the [[144,12,12]][[144,12,12]][[144,12,12]] gross code, can be viewed as cover codes. We also apply our method to search for BB codes with weight 8 checks and find many new codes, including a [[64,14,8]][[64,14,8]][[64,14,8]] and [[144,14,14]][[144,14,14]][[144,14,14]] code. For an hhh-cover code of an [[n,k,d]][[n,k,d]][[n,k,d]] BB code with parameters [[nh=hn,kh,dh]][[n_h = hn, k_h, d_h]][[nh​=hn,kh​,dh​]], we prove that kh≥kk_h \geq kkh​≥k for any hhh, and dh≤hdd_h \leq hddh​≤hd when hhh is odd. Furthermore if hhh is odd and kh=kk_h = kkh​=k, we prove the distance lower bound d≤dhd \leq d_hd≤dh​. We conjecture it is always true that an hhh-cover BB code of a base [[n,k,d]][[n,k,d]][[n,k,d]] BB code obeys the distance bounds d≤dh≤hdd \leq d_h \leq hdd≤dh​≤hd. While the focus of this work is on bivariate bicycle codes, we expect these methods to generalise readily to many group algebra codes and to certain code constructions involving hypergraph, lifted, and balanced products.
We study finite-length qudit quantum low-density parity-check (LDPC) codes from translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. Recent qubit work [PRX Quantum 6, 020357 (2025)] showed that, within the bivariate-bicycle viewpoint, twisting generalized toric patterns can significantly improve the finite-size distance of qubit codes. Building on this insight, we extend the construction and the search to qudit codes over finite fields. Using algebraic methods based on Laurent-polynomial ideals and Gröbner-basis computations over Fq\mathbb{F}_qFq​, we compute the number of logical qudits on any twisted torus and identify compact codes with favorable rate--distance tradeoffs. Overall, for the finite sizes explored, twisted-torus qudit constructions typically achieve larger estimated distances than their untwisted counterparts and, at the same number of physical and logical qudits, larger estimated distances than previously reported twisted qubit instances. The best new codes are tabulated.
We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a [[140,6,14]][[140,6,14]][[140,6,14]] code with kd2/n=8.40kd^2/n = 8.40kd2/n=8.40. In the code-capacity setting, the [[140,6,14]][[140,6,14]][[140,6,14]] code achieves a pseudothreshold of 8.0%8.0\%8.0% and kd2/n=8.40kd^2/n = 8.40kd2/n=8.40, exceeding the best multivariate bicycle code of Voss et al.\ (7.9%7.9\%7.9%, kd2/n=2.67kd^2/n = 2.67kd2/n=2.67). With circuit-level depolarizing noise, pseudothresholds reach 0.59%0.59\%0.59% for [[140,6,14]][[140,6,14]][[140,6,14]] and 0.53%0.53\%0.53% for [[84,6,10]][[84,6,10]][[84,6,10]]. On the SI1000 superconducting noise model, the [[140,6,14]][[140,6,14]][[140,6,14]] code achieves a per-round per-observable rate of 5.6×10−55.6 \times 10^{-5}5.6×10−5 at p=0.20%p = 0.20\%p=0.20%. We additionally present two self-dual codes with weight-8 stabilizers: [[54,14,5]][[54,14,5]][[54,14,5]] (kd2/n=6.48kd^2/n = 6.48kd2/n=6.48) and [[128,20,8]][[128,20,8]][[128,20,8]] (kd2/n=10.0kd^2/n = 10.0kd2/n=10.0). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.
Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the ZZZ-logical quotient K/SK/SK/S over F2[x,y]/(xℓ−1,ym−1)\mathbb F_2[x,y]/(x^\ell-1,y^m-1)F2​[x,y]/(xℓ−1,ym−1) and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity dZ=min⁡(dA,dC)d_Z=\min(d_{\mathcal A},d_{\mathcal C})dZ​=min(dA​,dC​). Using the Frobenius structure of the finite group algebra, we prove rA=rC=k/2r_{\mathcal A}=r_{\mathcal C}=k/2rA​=rC​=k/2 for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances 4,6,10,10,12,184,6,10,10,12,184,6,10,10,12,18 of the six standard BB codes of lengths 181818 to 288288288 and enumerates every minimum-weight logical operator. The resulting census shows that [[108,8,10]][[108,8,10]][[108,8,10]] is the only one of the six whose distance is attained in a single component, with dC=10d_{\mathcal C}=10dC​=10 and dA=12d_{\mathcal A}=12dA​=12.