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Singleton-Optimal Rank-Metric CSS Codes : Equality Structure and Exact Projected-Recovery Radii

Myeongjun Kim, Suseong Lee, Jaeho Jeon, Young-Sik Kim

arXiv:2609.36465v1cs.ITmath.IT

TL;DR

Shared-control faults can be physically dense yet low rank, motivating rank-metric CSS codes for stacked architectures. The paper proves bounds and characterizes equality and projected-recovery conditions, including when relaxing the recovery target does—and does not—extend the radius.

Abstract

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Correlated faults from shared control can have dense physical support yet low rank over a base field, motivating rank-metric quantum codes for stacked architectures. We prove an asymmetric rank-metric Singleton bound for Calderbank--Shor--Steane codes, including degenerate codes, and characterize equality through commuting maximum-rank-distance matrix codes. Optimal pairs exist for every admissible parameter triple and are necessarily pure. Comparison with erasure bounds establishes an exact advantage in stacked rank distance for general stabilizer codes at fixed physical resources on certain tall layouts. We determine whether relaxing the recovery target enlarges the worst-case correctable radius. For Singleton-optimal pairs with positive logical dimension, the measured syndrome determines the projected error modulo stabilizers exactly at radii strictly below half the sector rank distance, on every layout and for every nonzero projector. The projected syndrome is recoverable at arbitrary radii exactly when the complementary check space is invariant under the adjoint projector; otherwise, the same radius limit applies. For nontrivial idempotents, this invariance requires more rows than columns or sector distance one. For trace-self-adjoint projectors, linear ambient projected-syndrome interfaces in both sectors exist exactly when the code splits as a tensor product across the projector. Designs placing the two check spaces in complementary projector images are necessarily one-sided whenever they encode logical information. Two explicit families realize the extreme cases of the equality structure, including a two-sided family with an efficient certifying decoder attaining the optimal unique-decoding radius.

I. INTRODUCTION

Shared-control faults can affect many carriers while occupying a low-dimensional error subspace, motivating rank-metric codes for stacked architectures. The paper develops bounds, equality structure, and projected-recovery results for rank-metric CSS codes, then fixes the trace-pairing and physical-label conventions used in the analysis.

I. INTRODUCTION

  • Shared-control faults can be dense in physical support yet rank one over the base field, motivating rank-metric codes for stacked and modular architectures.The paper distinguishes this common-mode fault model from spatially extended radiation bursts, which need not be low rank.Source: §I · p. 1

A. Trace pairing and trace-dual spaces

I. INTRODUCTION

  • The central recovery question is whether projecting the target enlarges the worst-case correctable radius; the paper finds different answers for projected syndromes and projected errors modulo stabilizers.The former can be recoverable at every radius in an invariance case, while the latter never exceeds half the sector distance.Source: §I · p. 1

A. Main results

  • Every positive-logical-dimension CSS pair obeys the asymmetric rank-metric Singleton bound, including degenerate codes, while general stabilizer codes can exceed it in stacked rank distance at fixed resources.The comparison is a distance–dimension separation, not a claim of hardware advantage.Source: A. Main results · p. 2
  • Equality holds exactly for CSS-compatible pairs of maximum-rank-distance matrix codes, and every admissible parameter triple is attainable with pure optimal pairs.Degenerate codes do not attain the bound.Source: A. Main results · p. 2
  • For nonzero componentwise maps, projected errors modulo stabilizers are recoverable exactly below half the sector rank distance; projected syndromes have an all-radius criterion based on adjoint-projector invariance.For nontrivial idempotents, that invariance requires a tall layout or sector distance one.Source: A. Main results · p. 2
  • For trace-self-adjoint projectors, linear ambient projected-syndrome interfaces in both sectors exist exactly when the code splits as a tensor product across the projector.Complementary projector-image designs are one-sided whenever they encode logical information.Source: A. Main results · p. 2
  • Two witness families realize extreme equality structures, including quantum Gabidulin codes with exact sector distances and a certifying decoder at the optimal unique-decoding radius.The second family realizes the one-sided case with exact parameters on admissible instances.Source: A. Main results · p. 2

C. Physical embedding and rank invariance

The paper specifies how rank-metric labels map to physical Pauli operators and establishes which quantities that embedding preserves. It then fixes rank-distance notation and the regimes in which Gabidulin codes attain their distance and decoding guarantees.

D. Rank metric and code notation

C. Physical embedding and rank invariance

  • A common invertible transformation of every symbol preserves rank-metric quantities, but applying different transformations across carriers can change rank.Thus these quantities are invariants of the aligned layout, not of the code alone; physical check weights and distances can depend on the embedding convention.Source: C. Physical embedding and rank… · p. 4
  • For every prime power, trace-orthogonal check spaces realize a CSS stabilizer group whose physical normalizer is the trace-dual space.For non-prime q, the measured Fq-valued syndrome requires more generators, without changing the stabilizer group.Source: C. Physical embedding and rank… · p. 4

E. Gabidulin codes and the two regimes

  • For n ≤ m, Gabidulin codes attain dR = n − k + 1 and uniquely decode through t = floor((dR − 1)/2), while the stated guarantees do not apply when n > m.The latter regime lacks n Fq-independent evaluation points.Source: E. Gabidulin codes and the two regimes · p. 5

F. Matrix-label CSS pairs and the stacked rank distance

The section formulates CSS codes using matrix labels and sector rank distances, then proves an asymmetric rank-metric Singleton bound with a support-based equality criterion. The bound applies to degenerate pairs and uses the two sector distances separately.

F. Matrix-label CSS pairs and the stacked rank distance

  • For an a × n matrix-label CSS pair, K = an − dim SX − dim SZ, and the stacked rank distance is the minimum of the two sector rank distances.A rank-one burst represents a common-mode pattern through one fixed intra-carrier direction.Source: F. Matrix-label CSS pairs and the… · pp. 5–6

III. AN ASYMMETRIC RANK-METRIC SINGLETON BOUND

  • The trace-orthogonal formulation inherits the bound through Gram reduction, including for restrictions to a projector branch.The branch has row dimension dimFq Im(e).Source: F. Matrix-label CSS pairs and the… · p. 5, §III · p. 6
  • Every CSS pair with K > 0 obeys K ≤ an − max(a, n)(dR_X + dR_Z − 2), including degenerate pairs.The proof first bounds dimensions on disjoint row supports, then uses transposition when a > n.Source: §III · p. 6
  • Equality holds exactly when the check-space intersections satisfy the stated support conditions for some, equivalently every, admissible disjoint row-set pair.The conditions are SX ∩ ker πB = SX ∩ UA and SZ ∩ ker πA = SZ ∩ UB.Source: §III · p. 6

A. Comparison with the erasure bounds

The section compares the CSS Singleton bound with erasure bounds for general stabilizer codes and shows an exact stacked-rank-distance separation at fixed resources on specified tall layouts. The separation concerns the distance–dimension trade-off, not hardware performance.

A. Comparison with the erasure bounds

  • For a stabilizer code on an a × n layout with K > 0 and stacked rank distance D, where D − 1 = 2κ + ι, the column bound is strictly smaller than the row bound exactly when κ ≥ 1 and a > 2n + 2ι.Source: A. Comparison with the erasure bounds · p. 7
  • The CSS bound’s improvement over the row erasure bound includes a gain from using sector distances separately; transposition adds a further gain when a > n and D ≥ 2.For a = n = 5, dR_X = 2, and dR_Z = 3, the erasure bound gives K ≤ 15 while the CSS bound gives K ≤ 10.Source: A. Comparison with the erasure bounds · pp. 7–8
  • A 3 × 2 stabilizer code over F2 has K = 2 and D = 2, although every positive-logical-dimension CSS pair on that layout has D = 1.The construction and the bound establish that the CSS Singleton bound does not extend to general stabilizer codes.Source: A. Comparison with the erasure bounds · p. 8
  • On the 2m × m layouts, the cited stabilizer family attains the row erasure bound, while the CSS bound can be strictly tighter; layout orientation affects its stacked rank distance.For m = 2 and k = 1, the code has K = 4 and D = 2 on 4 × 2, but D = 1 on the transposed 2 × 4 layout.Source: A. Comparison with the erasure bounds · p. 8
  • At fixed layout, physical-qubit count, and logical dimension, the maximum stacked rank distance is k + 1 for stabilizer codes versus ⌊k/2⌋ + 1 for CSS pairs.The distance gap is exactly ⌈k/2⌉; for (a, n, K, D) = (12, 3, 12, 3), the row bound allows K ≤ 24 while the column bound allows K ≤ 12.Source: A. Comparison with the erasure bounds · p. 9
  • The separation is not a hardware-advantage claim: fixed physical resources and layout do not fix check weights, extraction depth, or gate error rates.The comparison isolates a distance–dimension trade-off.Source: A. Comparison with the erasure bounds · p. 9

IV. THE STRUCTURE OF SINGLETON-OPTIMAL PAIRS

Singleton equality rigidly characterizes the CSS check spaces and the attainable parameters. Every admissible parameter triple is realized, and optimal pairs with positive logical dimension are necessarily pure.

IV. THE STRUCTURE OF SINGLETON-OPTIMAL PAIRS

  • Equality in the asymmetric Singleton bound forces each nonzero check space to be a maximum-rank-distance matrix code with the corresponding sector distance.The equality structure also requires the check-space support-intersection conditions.Source: §IV · pp. 9–10
  • For u, v ≥ 0 with u + v ≤ L − 1, Singleton-optimal pairs exist over every finite field and layout, with K = M(L − u − v).The construction uses nested Gabidulin codes expanded in trace-dual bases.Source: §IV · p. 10
  • The nonzero optimal check spaces have dimensions Mv and Mu and sector distances L − v + 1 and L − u + 1, respectively.These MRD parameters yield sector distances dR_X = u + 1 and dR_Z = v + 1 for the pair.Source: §IV · pp. 10–11
  • Every Singleton-optimal pair with K > 0 is pure; degeneracy is therefore excluded by optimality, not assumed away.A degenerate code with a stabilizer label below its sector distance cannot attain the bound.Source: §IV · p. 11

V. PROJECTED-RECOVERY INTERFACES AND THEIR EXACT RADII

This section asks whether weaker projected-recovery targets enlarge the correctable radius when rank-metric decoders cannot generally obtain a projected syndrome by projecting a measured syndrome. It defines six interface conditions and reports strictness of all one-way implications.

V. PROJECTED-RECOVERY INTERFACES AND THEIR EXACT RADII

  • Because rank-metric decoding operators are generally only Fq-linear, projecting a globally measured syndrome does not generally supply the syndrome for a projected error component.The section studies recovery of either the projected syndrome or the projected error modulo stabilizer labels.Source: §V · p. 12
  • The six interface conditions distinguish whole-space from rank-ball recovery, linear from arbitrary-function maps, and projected-syndrome from projected-error-modulo-stabilizer targets.The definitions are given for each Pauli sector.Source: §V · p. 12
  • Every one-way implication among the interface conditions is strict, with witnesses listed in Table I.The section states that it establishes exact criteria and values for Singleton-optimal pairs.Source: §V · p. 12

A. Ambient and relative factorization

The factorization criteria characterize when a measured trace syndrome can determine a projected syndrome, both on componentwise subspaces and on the full label space. Relative recovery allows a correction from the trace-orthogonal complement, while ambient recovery requires invariance of the check span under the adjoint projector.

A. Ambient and relative factorization

  • On W = U^n, projected-syndrome recovery is equivalent to the measured syndrome factoring through the projection on W, with check-span condition e† ⋆ S_H ⊆ S_H + W⊥Tr.The factor map may be chosen linear; it may be the identity exactly when each e† ⋆ h(i) − h(i) lies in W⊥Tr.Source: A. Ambient and relative factorization · p. 13
  • For the full label space, W⊥Tr = {0}, so projected-syndrome recovery is equivalent to e† ⋆ S_H ⊆ S_H.Identity recovery requires every check label to be fixed by e†.Source: A. Ambient and relative factorization · p. 13
  • The relative criterion has no gap between arbitrary and linear recovery maps, because a linear map on the image factors linearly through the measured syndrome.The correction term W⊥Tr cannot generally be omitted: the supplied example satisfies relative identity recovery despite e† ⋆ h − h being nonzero.Source: A. Ambient and relative factorization · p. 13

B. Bounded-rank compatibility and certified decoding

Bounded-rank recovery is governed by the minimum rank in the syndrome kernel: below half that distance, rank-t errors have unique syndromes and can be certified. A rank-t linear compatibility identity extends to all errors in the componentwise subspace.

B. Bounded-rank compatibility and certified decoding

  • A kernel element of rank at most 2t in W can be written as a difference of two elements of rank at most t.This rank-halving property supports the uniqueness and bounded-rank factorization criteria.Source: B. Bounded-rank compatibility and… · pp. 13–14
  • A certifying radius-t decoder returns the unique rank-at-most-t syndrome preimage, or ⊥ otherwise; membership, rank, and syndrome checks validate a candidate.Those checks alone do not rule out a second preimage; uniqueness follows a priori when d_R(K) > 2t.Source: B. Bounded-rank compatibility and… · p. 14
  • d_R(K) > 2t exactly characterizes when every syndrome has at most one preimage in W of rank at most t.Under this condition, the certifying decoder returns that unique error for every rank-at-most-t input.Source: B. Bounded-rank compatibility and… · p. 14
  • When d_R(S_H⊥Tr) > 2t, every Fq-linear projector and check family admits a projected-syndrome interface on the rank-t ball.The stated construction uses a certifying radius-t decoder, at the cost of one bounded-rank decoding per syndrome.Source: B. Bounded-rank compatibility and… · p. 14
  • If a linear projected-syndrome interface exists for all errors of rank at most t in W = U^n, the same identity holds throughout W.For ambient W, the resulting check-span condition is e† ⋆ S_H ⊆ S_H; with identity interface, each check is fixed by e†.Source: B. Bounded-rank compatibility and… · p. 14

C. Recovering the projected error modulo stabilizer labels

Projected-error recovery modulo a check space is characterized by whether projection maps syndrome-kernel errors into that space. This target is stronger than projected-syndrome recovery but weaker than recovering the projected error itself.

C. Recovering the projected error modulo stabilizer labels

  • A T-valued projected interface exists exactly when e ⋆ ker σ_H ⊆ T; at T = S_X, this target is strictly stronger than projected-syndrome recovery and weaker than exact projected-error recovery.The minimum-rank certificate gives a sufficient bounded-rank interface condition for every T.Source: C. Recovering the projected error… · p. 15
  • A linear quotient-valued interface on a rank-t ball extends to all of W, while interfaces outside the resulting kernel-containment criterion are nonlinear.For trace-self-adjoint idempotents at T = S_X, the criterion matches the stated structural characterization; this equivalence need not hold for general linear maps.Source: C. Recovering the projected error… · p. 15
  • For the supplied rank-one idempotent example, a zero linear projected-syndrome interface exists, yet errors of rank at most one do not have recoverable projected errors modulo S_X.The example separates projected-syndrome recovery from projected-error recovery modulo stabilizers.Source: C. Recovering the projected error… · p. 15
  • The class with both check spans invariant under the projector is distinct from the class with quotient-valued projected-error recovery.Conflating these linear classes would miss the second structural condition and its dimensional cost.Source: C. Recovering the projected error… · p. 15

D. The two relative distances and the exact radii

The two relative distances give exact tests for bounded-rank syndrome and quotient-valued interfaces. On Singleton-optimal pairs, the quotient-valued target has the sector-distance radius limit, while projected-syndrome recovery can be unrestricted when an adjoint-invariance condition holds.

D. The two relative distances and the exact radii

  • For every t ≥1, I5,X(t) holds exactly when 2t < δsyn(e), and I6,X(t) exactly when 2t < δquot(e).The invariants satisfy dR(D) ≤ δquot(e) ≤ δsyn(e).Source: D. The two relative distances and the… · p. 16
  • On any layout, δsyn(e) is +∞ exactly when e ⋆DX ⊆DX, equivalently e† ⋆SZ ⊆SZ; otherwise it equals dR X.The complementary-check-space invariance condition determines whether projected-syndrome recovery can hold at arbitrary radii.Source: D. The two relative distances and the… · p. 16
  • In the X sector, for every t ≥ 1, I6,X(t) holds if and only if 2t < dR^X; thus the stated strict threshold applies only to t values at least 1.Source: D. The two relative distances and the… · p. 16
  • For m ≤ n and nontrivial idempotents, if both sector distances are at least two, neither check space is invariant under e or e†, precluding an ambient projected-syndrome interface.The one-sided endpoint dR X = 1 is an exception: δsyn(e) can be +∞ while δquot(e) = 1.Source: D. The two relative distances and the… · pp. 16–17
  • The quotient-valued target Fe(E) = e ⋆E + SX need not be determined by a stabilizer class unless e ⋆SX ⊆SX.Thus recovering this target is not by itself recovery of an independent logical subsystem; the residual component requires a distinct recovery mechanism.Source: D. The two relative distances and the… · p. 17

VI. LINEAR INTERFACES: SPLITTING, LOGICAL COST AND ONE-SIDEDNESS

Linear interfaces impose structure on the code: interfaces in both sectors correspond to a tensor split across a trace-self-adjoint projector. Aligned checks can supply projected syndromes directly, and self-adjoint idempotents exist at every rank.

A. Compatible check families split

  • With a trace-self-adjoint idempotent, linear ambient projected-syndrome interfaces in both sectors exist exactly when the code is a tensor product across the projector split.A one-sector interface alone does not force the other check space to split.Source: A. Compatible check families split · p. 18

B. Aligned families and trace-self-adjoint idempotents

  • For self-adjoint e, measured syndromes equal projected syndromes check by check exactly when the check family lies in Fix(e)^n.If e is idempotent, projecting an arbitrary check family produces aligned checks, though some may vanish or become dependent.Source: B. Aligned families and… · p. 18
  • Complementary images of a self-adjoint idempotent are trace-orthogonal, making check families placed in those images CSS-admissible.The two image spaces have dimensions r and m − r.Source: B. Aligned families and… · pp. 18–19

C. Existence without basis hypotheses

  • For every prime power q, extension degree m, and rank 0 ≤ r ≤ m, a self-adjoint Fq-linear idempotent of rank r exists.Equivalently, the trace bilinear space admits nondegenerate subspaces of every dimension.Source: C. Existence without basis hypotheses · p. 19

D. The trace-compatible CSS split

Complementary images of a trace-self-adjoint idempotent provide trace-orthogonal spaces for CSS checks. Ambient quotient-valued interfaces require both check spans to be invariant and the projected branch to carry no logical qudit; adding checks to achieve one costs exactly that branch’s logical dimension.

D. The trace-compatible CSS split

E. The logical cost of an ambient quotient-valued interface

  • An ambient quotient-valued interface exists exactly when both check spans are e-invariant and the Im(e) branch carries no logical qudit.When Kq > 0 under this condition, all logical operators are supported on the ker(e) branch.Source: E. The logical cost of an ambient… · p. 20
  • Creating an ambient quotient-valued interface by adding X-checks costs at least KIm(e) independent generators and reduces the logical dimension by exactly KIm(e).The minimum is attained by the specified enlargement of SX.Source: E. The logical cost of an ambient… · p. 20

F. One-sidedness of trace-aligned complementary designs

Trace-aligned complementary designs cannot protect both Pauli sectors in rank distance when they encode logical information. The two witness families realize opposite equality-structure endpoints, with the two-sided quantum Gabidulin family supporting efficient certified decoding at the optimal radius.

F. One-sidedness of trace-aligned complementary designs

  • If both check-space containments are proper, then both sector rank distances equal one, so positive-logical-dimension complementary designs are necessarily one-sided.The theorem applies for any q, m, n, r, and choice of check spaces.Source: F. One-sidedness of trace-aligned… · p. 21

G. Structure of the saturated codes

  • When SX = V^n, the realized code factors into frozen |+⟩ directions and a residual code on the remaining qudits, whose Z-rank distance is one whenever it encodes logical information.The frozen directions carry no logical information, so they provide no logical-Z protection.Source: G. Structure of the saturated codes · p. 21

VII. TWO WITNESS FAMILIES

  • The two witness families realize the equality-structure endpoints: quantum Gabidulin codes are two-sided, while projected Gabidulin codes realize the one-sided endpoint.The two-sided family also has an efficient certifying decoder at the optimal unique-decoding radius.Source: §VII · p. 22, A. Two-sided: quantum Gabidulin codes · p. 23

A. Two-sided: quantum Gabidulin codes

  • For QGab(α, r, r), both logical sector distances equal r + 1, and neither check span is invariant under any admissible nontrivial idempotent.For every t ≤ floor(r/2), both sectors admit minimum-rank certificates, but none is given by an F2-linear map on the ambient label space.Source: A. Two-sided: quantum Gabidulin codes · p. 22
  • The quantum Gabidulin family has a certifying decoder for every t ≤ floor(r/2), the largest possible unique-decoding radius.The procedure reduces the measured binary trace syndrome to Gabidulin syndrome decoding with certification; it is not a new rank-decoding algorithm.Source: A. Two-sided: quantum Gabidulin codes · p. 23

B. One-sided: idempotent-generated Gabidulin codes

In the square aligned regime, idempotent-generated Gabidulin codes yield a one-sided Singleton-optimal family. Saturation determines the protected branch's logical distance, while the construction has explicit scope limits.

B. One-sided: idempotent-generated Gabidulin codes

  • The construction assumes n = m with independent evaluation points; the projected-dimension formula need not hold when n < m and the evaluation construction does not extend to n > m.For n > m, dependent points yield a rank-one label that obstructs this specific construction, not rank-metric CSS codes in general.Source: B. One-sided: idempotent-generated… · pp. 24, 26
  • For the saturated construction, the protected branch is MRD and its restricted trace dual has distance n − kZ + 1.The MRD assertion uses r < kZ < m; the theorem also gives endpoint cases separately.Source: B. One-sided: idempotent-generated… · p. 25
  • The saturation hypothesis is essential: without it, restricted-dual distance need not coincide with logical distance; for q = 2, m = n = 4, r = 2, and kX = kZ = 3, dR(S_Z^⊥) = 2 = m − kZ + 1.Source: B. One-sided: idempotent-generated… · p. 25
  • In the square aligned regime, the saturated branch code has Kq = m(kZ − r), Z-rank distance n − kZ + 1, and X-rank distance one.It attains the Singleton bound for every admissible (m, r, kZ) in the stated regime.Source: B. One-sided: idempotent-generated… · p. 25

A. Relation to prior work

The paper extends rank-metric Singleton and equality analyses beyond their Hamming-metric counterparts, then specifies the correlated-fault regime it targets and boundaries of its results. It reports exhaustive finite checks and identifies open questions, including fault-tolerant syndrome extraction.

A. Relation to prior work

  • Rank-metric Singleton analysis uses row-support shortening and transposition, and its equality statements are unconditional, with purity a conclusion rather than an assumption.The paper also identifies its two relative distances with first relative generalized matrix weights while emphasizing the projector-induced nested pairs and their exact evaluation.Source: A. Relation to prior work · p. 26

B. Intended regime and limitations

  • The intended regime is correlated faults aligned along intra-carrier directions; the paper makes no claim of uniform superiority over baseline codes in arbitrary noise environments.Measurement faults, high-weight-check overhead, and circuit-level implications of one-sidedness are not analyzed.Source: B. Intended regime and limitations · p. 27

C. Finite verifications and reproducibility

  • Finite verifications include exhaustive enumeration of 92,881 trace-orthogonal pairs with positive logical dimension, of which 97 attain the bound.The reported scripts and field and basis specifications are supplied as supplementary material.Source: C. Finite verifications and… · p. 27

D. Conclusion and open problems

APPENDIX A REALIZATION OVER A NONPRIME BASE FIELD

The appendices address syndrome measurement over nonprime base fields and give a stacked construction with an invariant projector. The latter provides an example where projected-syndrome recovery has no radius restriction despite nontrivial logical content in the projector branch.

APPENDIX A REALIZATION OVER A NONPRIME BASE FIELD

  • Over q = p^ν, ν measured trace parities determine the Fq-valued syndrome, increasing the measured generators ν-fold without changing the stabilizer group.The outcomes are converted through a fixed offline Fp-linear map.Source: Appendix A · pp. 27–28

APPENDIX B A SINGLETON-OPTIMAL PAIR CARRYING AN INVARIANT PROJECTOR

  • Stacking square Singleton-optimal pairs on disjoint row blocks yields a tall Singleton-optimal pair invariant under each block projector.When a > n, both sectors have exact Fq-linear ambient projected-syndrome interfaces at every radius, although the projector branch carries logical qudits.Source: Appendix B · p. 28
  • For the stacked construction, the projector's syndrome is a linear function of the full syndrome, but no syndrome function returns the projected error modulo stabilizers in the branch carrying logical information.The appendix gives block-projector examples with δsyn(Pi) = +∞ while δquot(Pi) = dR_X.Source: Appendix B · p. 28
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