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Coset Lifted Product Codes: Constructing Quantum LDPC Codes from Non-Normal Subgroup Actions

Rajendra Prasad Rajpurohit, V. Lalitha, Shayan Srinivasa Garani

arXiv:2610.12782v1quant-phcs.ITmath.IT

TL;DR

Short-block lifted-product codes typically use regular group actions, while coset actions allow non-normal subgroups but had been used in other constructions. This paper combines coset actions with protographs to define CLP CSS codes and derives an odd-order dimension formula. Its search reports codes exceeding the best published Q values at check weights 6, 7, and 9.

  • Problem

    Short-block lifted-product codes use the regular representation, while prior coset-action constructions provide a distinct route involving non-normal subgroups.

    Source: §I · p. 2

  • Method

    CLP codes apply protographs through left and right actions on G/H, and their odd-order dimension formula uses irreducible representations and subgroup fixed spaces.

    Source: §I · p. 2, A. Coset lifted product codes · p. 6, B. Dimension · p. 8

  • Results

    CLP codes exceed reported best Q values at w=6, 7, and 9; at w=7 they attain Q=26.67 versus 24.00 for the bound-based comparator.

    Source: §IV · p. 10

  • Takeaways & Limitations

    The CLP framework includes lifted products, two-block group algebra codes, and coset-based two-block codes as special cases.

    Source: A. Coset lifted product codes · p. 6, §V · p. 12

    The candidate at w=10 has distances that remain to be certified.

    Source: §V · p. 12

Abstract

from arXiv · show

We introduce coset lifted product (CLP) codes, a family of quantum LDPC codes that combine the protograph structure of lifted product codes with the coset action of Aydin, Tamo, and Barg \cite{AydinTamoBarg2026}. Given a finite group $G$, a subgroup $H \le G$ that need not be normal, a protograph $A$ over $\mathbb F_2[G]$, and a protograph $B$ over $\mathbb F_2[N_G(H)]$, the two protographs act on the cosets $G/H$ from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations $\rho$ of $G$ of hypergraph product dimensions. A search over $14,843$ pairs $(G,H)$ with $H$ restricted to be non-normal and $120 \le n \le 400$ yields certified codes that exceed the best published value of $Q=kd^2/n$ at check weights $6,7$ and $9$, i.e., we obtain $[[360,20,20]]$ at $w=6$, a $[[360,24,20]]$ code at $w=7$, and codes $[[252,16,18]]$ and $[[288,16,20]]$ at $w=9$. Our search also produce high-distance codes with parameters $[[168,10,14]], [[240,8,20]]$ at $w=6$ and $[[216,8,18]]$ at $w=7$.

I. INTRODUCTION

The introduction motivates sparse quantum codes by the rate and distance limits of earlier constructions, then places CLP codes among approaches using group actions on cosets. The preliminaries establish CSS, LDPC, and coset-action notation used by the construction.

A. Classical and quantum codes

  • For CSS codes, orthogonal check matrices make the X- and Z-type stabilizers commute; quantum LDPC checks have bounded row and column weights, and the existence of good families had remained open for nearly two decades.The CSS condition turns code design into finding mutually orthogonal classical check matrices.Source: §I · pp. 1–2, A. Classical and quantum codes · p. 3

I. INTRODUCTION

  • Surface codes require n = Θ(d^2) qubits to protect one logical qubit, while the hypergraph product has distance d = Θ(√n), motivating quantum LDPC constructions with stronger scaling.A constant-rate quantum LDPC family with an efficient decoder and constant noise threshold permits constant space overhead; good families additionally have linear distance.Source: §I · pp. 1–2
  • Earlier lifted-product constructions over cyclic groups lacked constant rate, while non-abelian lifted products established asymptotically good quantum LDPC codes and settled the qLDPC conjecture.Panteleev and Kalachev, obtained dimension Θ(log n) and distance Θ(n / log n) in a random cyclic-cover setting; their later non-abelian construction achieved constant rate and linear distance under expansion conditions.Source: §I · p. 2
  • At short block lengths, 2BGA codes use left and right regular representations, while Aydin, Tamo, and Barg [25] use commuting actions on G/H, even for non-normal H, yielding two-block codes outside 2BGA.Source: §I · p. 2

B. Group actions and group algebras

  • On cosets G/H, the left action maps g to L(g), while the right action maps n in the normalizer N to R(n); their group-algebra extensions act linearly on matrices.The right action reverses product order, R(nn′) = R(n′)R(n), while L(gg′) = L(g)L(g′).Source: B. Group actions and group algebras · p. 4

C. Group algebra quantum LDPC codes

CLP codes apply protograph matrices through left and right actions on cosets, producing CSS codes even when the subgroup is non-normal. The construction includes lifted products and coset-based two-block codes as special cases.

A. Coset lifted product codes

  • CLP(A, B; G, H) applies A over F2[G] from the left and B over F2[N_G(H)] from the right on cosets G/H, using the resulting block matrices to form a CSS code.The block construction has n = m(nA1nB0 + nA0nB1) qubits, where m = |G/H|.Source: A. Coset lifted product codes · p. 5
  • The left and right coset actions commute, so the CLP check matrices satisfy CSS orthogonality for every finite G and subgroup H, normal or not.The proof pairs the two block-product terms and uses their equality to cancel over F2.Source: A. Coset lifted product codes · p. 5
  • When H is trivial, CLP is a lifted product over the regular representations; with 1 × 1 protographs it is a coset-based two-block code, and normal H reduces it to a lifted product over G/H.Every CLP code can also be represented using a core-free subgroup, by passing to G/CoreG(H).Source: A. Coset lifted product codes · p. 6
  • In the A4 example, H is a non-normal subgroup of order 2; with m = 6, the construction yields a [[48, 6, 4]] code with dX = dZ = 4.Gaussian elimination gives rank HX = rank HZ = 21, so k = 6.Source: A. Coset lifted product codes · p. 6
  • Equivalent protographs can yield equivalent CLP codes under the stated entry changes and group automorphisms, preserving code parameters and check weights.The equivalence is implemented by permutations of qubits and checks.Source: A. Coset lifted product codes · p. 7

B. Dimension

For groups of odd order, the dimension analysis extends to CLP codes with arbitrary subgroups, using irreducible representations and subgroup fixed spaces. This replaces the abelian-case decomposition into finite fields and accommodates coset actions that are not free.

B. Dimension

  • For every odd-order group and subgroup, the CLP dimension is expressed using irreducible representations, with the subgroup entering through its fixed-space dimension in each representation.The resulting matrices have sizes dim ρ and dim ρ^H; this differs from the homology-of-factors formula for balanced products because a coset action is not free.Source: B. Dimension · p. 8
  • The representation-based formula extends the dimension treatment beyond the abelian odd-order lifted-product case, where the group algebra splits into finite fields and the dimension sums hypergraph-product dimensions.No general closed formula for lifted-product dimension is known in the cited discussion.Source: B. Dimension · p. 8

1) Preliminaries from representation theory:

For odd-order groups, the coset representation can be decomposed into irreducible representation blocks, yielding a dimension formula for CLP codes as a sum of hypergraph product dimensions. The subgroup enters through the fixed spaces in those representations.

1) Preliminaries from representation theory:

  • The coset action decomposes into irreducible blocks, with each ρ appearing µρ times and Σρ dρµρ = m.The multiplicity µρ is the dimension of the space fixed by H in the representation.Source: 1) Preliminaries from representation… · p. 8
  • For |G| odd, the CLP dimension equals the sum of the dimensions of HGP(Aρ, Bρ) over irreducible representations with µρ > 0.The paper equivalently identifies this dimension with that of the direct sum of those hypergraph product codes.Source: 1) Preliminaries from representation… · p. 9
  • The dimension formula follows by changing basis to representation blocks, where the X- and Z-check matrices become the corresponding HGP check matrices.Ranks are preserved under the changes of basis and permutations, and the block lengths sum to the CLP length.Source: 1) Preliminaries from representation… · p. 9
  • For β in F2[N], τρ(β) is obtained from the action of N on the H-fixed row-vector space Wρ, and τρ(nn′) = τρ(n′)τρ(n).This gives an explicit way to compute the representation blocks without computing the change-of-basis matrix S.Source: 1) Preliminaries from representation… · p. 9
  • For the stated A and B, n = 45: direct elimination gives rank HX = 17 and rank HZ = 18, while Theorem III.7 gives k = 10 term by term and Formula (24)'s contributions sum to 10.Source: 1) Preliminaries from representation… · p. 10

IV. SEARCH FOR CODES

The search reports certified CLP codes that exceed published Q values at check weights 6, 7, and 9. A weight-10 candidate also exceeds a certified comparison, but its distance remains an upper bound.

IV. SEARCH FOR CODES

  • At w = 7, [[360, 24, 20]] reaches Q = 26.67, above 18.00 for [[288, 16, 18]] and 24.00 for [[384, 16, ≤24]].The second comparison is bound-based.Source: §IV · p. 10
  • At w = 6, [[360, 20, 20]] reaches Q = 22.22, above 15.25 for [[340, 16, 18]] and 13.71 for [[224, 12, 16]].The first is identified as the leading exact generalized bicycle code; the second is the code of [25].Source: §IV · p. 10
  • At w = 9, [[252, 16, 18]] reaches Q = 20.57, above 20.25 for [[288, 18, 18]].Source: §IV · p. 10
  • At w = 10, candidate [[360, 20, ≤27]] reaches Q ≤40.50, above the certified 38.77 but below the bound-based endpoint.The candidate's distance remains an upper bound.Source: §IV · p. 10

A. Conventions and search space

The search narrows the CLP design space, screens candidates against check-weight-specific Q thresholds, and certifies surviving distances. Its certified codes exceed published figures at weights 6, 7, and 9; the weight-10 result remains a candidate.

A. Conventions and search space

  • The main search covered 14,843 pairs with 30 ≤ m ≤ 100 (120 ≤ n ≤ 400), under subgroup, index, support, and entry-weight restrictions, and was restricted to 1 × 1 over 2 × 2.That layout used a single element a and a 2 × 2 matrix B.Source: A. Conventions and search space · p. 10

B. Pipeline and certification

  • Draws were screened using QDistRnd estimates on both sides, then survivors were rescreened at 10^6 rounds; six of seven high-bound candidates lost 1–5 distance.Bounds above 25 should therefore be read with care.Source: B. Pipeline and certification · p. 11
  • Distances were certified by binary integer programs; a sector with no solution below the cutoff supports a lower bound, while an exact-weight witness pins the distance.A separate script rechecks orthogonality, dimension, check weights, and witnesses.Source: B. Pipeline and certification · p. 11

C. Certified CLP codes and candidates search results

V. CONCLUSION

  • Certified CLP codes exceed the best published figure of merit at check weights 6, 7, and 9; the w = 10 candidate still needs distance certification.Source: §V · p. 12

APPENDIX A PROTOGRAPHS OF THE CODES

The appendix lists protographs and group data for certified codes and candidates. The reported codes include highlighted examples at check weights 6, 7, and 9.

a) Certified codes, 1 × 1 over 2 × 2

  • [[168, 10, 14]] at w = 6 and [[240, 8, 20]] at w = 6 are additional certified codes.The appendix identifies their groups, subgroups, and, where known, normalizers.Source: Appendix A · pp. 13–14
  • [[360, 20, 20]] at w = 6, [[360, 24, 20]] at w = 7, and [[252, 16, 18]] and [[288, 16, 20]] at w = 9 are highlighted codes.The entries also give the groups and subgroups used in each construction.Source: Appendix A · p. 14
  • Other listed w = 7 codes include [[168, 14, 12]], [[180, 16, 12]], [[224, 24, 9]], and [[252, 22, 10]].The appendix gives the group and subgroup for each entry.Source: Appendix A · pp. 13–14

b) Certified codes, 2 × 2 over 1 × 1

  • The appendix also lists certified codes at w = 5, including [[120, 8, 8]], [[180, 12, 8]], and [[240, 16, 8]].Each listed code is accompanied by group and subgroup data.Source: Appendix A · pp. 13, 15

a) Certified codes, 1 × 1 over 2 × 2

  • [[216, 8, 18]] at w = 7 is another certified code.Its entry specifies G = SmallGroup(162, 3) and H = ⟨f3f5⟩.Source: Appendix A · p. 15

c) Candidates

  • The candidate list includes w = 9 codes with parameters [[224, 12, ≤21]], [[288, 12, ≤23]], and [[288, 12, ≤24]].The section states that these candidates have a 106-round upper bound on both sides.Source: Appendix A · p. 15

APPENDIX B SMALLGROUP(l, i) STRUCTURES

The appendix describes the group structures appearing in Tables I and II and clarifies how to interpret their labels.

APPENDIX B SMALLGROUP(l, i) STRUCTURES

  • The table reports group structures as given by GAP’s StructureDescription.D2n denotes the dihedral group of order 2n; Q8 and Q16 denote quaternion groups of orders 8 and 16.Source: Table III · p. 16
  • A group name is only a gloss; the SmallGroups identifier and polycyclic presentation define the group.Source: Table III · p. 16
  • The table covers groups appearing in Tables I and II.Source: Table III · p. 16

APPENDIX C A WORKED EXAMPLE OF THE CODE CONSTRUCTION AND THEOREM III.7

The appendix works through a small odd-order, nonabelian example to illustrate the code construction and Theorem III.7. It decomposes the checks by irreducible representations and verifies that the resulting code has dimension 10.

a) Step 0. The data of Definition III.1.

  • The worked example uses a nonabelian group of order 27 and a non-normal subgroup H = {1, x, x2}.The text establishes non-normality using yxy−1 = xz−1, which is not in H.Source: Appendix C · p. 16
  • The nine left cosets form a set Ω, so m = |Ω| = 9; the example uses a 2 × 2 choice for each protograph.The resulting code length is n = 9 · 8 = 72.Source: Appendix C · p. 16

c) Step 2. The irreducible representations and the use of (F2).

  • The representation calculation identifies five representations in F[G/H]: three one-dimensional representations and the pair ρ, ¯ρ.Their multiplicities are each 1.Source: Appendix C · p. 17

e) Step 4. The change of basis S and the use of (F1).

  • A change of basis block-diagonalizes the left actions, while the commuting right actions decompose into representation blocks through Schur’s lemma.The example then constructs the corresponding Aρ and Bρ matrices.Source: Appendix C · p. 18

h) Step 7. Step 2 of the proof, the X checks and Matrix Permutation.

  • The five X-check blocks have ranks 3, 4, 4, 10, and 10, giving rank HX = 31; the Z-check rank is also 31.Direct elimination over F2 confirms both ranks and k = 72 − 62 = 10.Source: Appendix C · p. 19

k) Step 10. The kernel form (28) through (24).

  • The representation-wise hypergraph product contributions sum to dimension 10, matching the direct check-matrix calculation.The example concludes that the CLP code is [[72, 10]] and has the same dimension as the direct sum of the five hypergraph product codes.Source: Appendix C · p. 19
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