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arxiv: 2605.23894 · v1 · pith:N3QP7NTPnew · submitted 2026-05-22 · 🪐 quant-ph · cs.IT· math.IT

A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases

Koki Okada , Kenta Kasai This is my paper

Pith reviewed 2026-05-25 03:59 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords CSS LDPC codesquantum error correctionfinite-field constructionbase matrixcyclic liftTanner graph girthbelief propagation decodingmultiplicative cosets
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The pith

A two-branch multiplicative-coset construction reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion for quantum LDPC bases to explicit finite-field quotient-coset conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a systematic algebraic method to build regular CSS quantum LDPC base matrices for chosen column weight J and even row weight L. It converts the three main design constraints into searchable conditions on cosets in a finite field, then uses normalized exhaustive search to find valid bases for several degree pairs. The base fixes the degree distribution and initial girth rules, after which a cyclic lift randomizes connections while enforcing exact algebraic checks. A sympathetic reader would care because the separation of stages yields concrete codes, such as a 64-fold lift of a (3,10) base that reaches girth at least eight in the same-type Tanner graphs and excludes a weight-16 logical orbit.

Core claim

The two-branch multiplicative-coset construction reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion to explicit quotient-coset conditions over a finite field. Normalized exhaustive search produces base matrices for several (J,L) pairs. A 64-fold cyclic lift of the (3,10) example yields a [[10240,4108,10≤d≤32]] CSS code whose same-type graphs have girth at least eight and excludes the specified weight-16 nondegenerate logical-support orbit. Joint log-domain belief propagation plus low-complexity post-processing for small residual syndromes achieves a post-processing frame error rate of 1.0×10^{-7} at depolarizing probability p=0.058.

What carries the argument

Two-branch multiplicative-coset construction over a finite field, which converts regularity, CSS orthogonality, and same-type 4-cycle exclusion into explicit quotient-coset conditions that exhaustive search can meet.

If this is right

  • Base matrices exist for multiple (J,L) pairs without tying the method to one degree distribution.
  • The 64-fold lift of the (3,10) base produces a code with girth at least eight in both same-type graphs and excludes the target logical-support orbit.
  • Joint belief-propagation decoding plus deterministic post-processing for residual syndromes with two unsatisfied checks yields a post-processing FER of 1.0×10^{-7} at p=0.058.
  • The two-stage separation allows the base to fix degree and girth constraints while the lift handles randomization under algebraic checks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same quotient-coset conditions could be solved in larger fields to generate bases for additional (J,L) pairs not yet enumerated.
  • Excluding specific logical-support orbits at the base stage may systematically improve the minimum distance bounds of the lifted codes.
  • The algebraic separation of base and lift stages offers a template for constructing other families of quantum LDPC codes that require both regularity and CSS orthogonality.

Load-bearing premise

The chosen finite field admits solutions to the quotient-coset conditions for the target (J,L) pairs and the cyclic lift preserves girth while excluding the specified logical orbits without introducing new short cycles or violating orthogonality.

What would settle it

An exhaustive search that returns no base matrix satisfying the quotient-coset conditions for a given (J,L) pair, or a lifted code whose same-type Tanner graph contains a 4-cycle or whose logical support includes the excluded weight-16 orbit.

Figures

Figures reproduced from arXiv: 2605.23894 by Kenta Kasai, Koki Okada.

Figure 1
Figure 1. Figure 1: FER of the 64-fold lift for the [[n, k, d]] = [[10240, 4108, 10 ≤ d ≤ 32]] CSS code. The hashing line is the depolarizing-channel quantum hashing bound for the effective rate R = 4108/10240. The DE line is an approximate BP density-evolution reference for the regular (3, 10) ensemble. In the p = 0.058 run, joint BP with post-processing produced 25 recorded failures. The run contained 180,000,000 trials. Th… view at source ↗
read the original abstract

This paper develops a two-branch multiplicative-coset construction for regular Calderbank-Shor-Steane (CSS) quantum low-density parity-check base matrices. For a target column weight \(J\) and an even row weight \(L\), the method reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion to explicit quotient-coset conditions over a finite field. A normalized exhaustive search for these conditions produces base matrices for several \((J,L)\) pairs, so the construction is not tied to a single degree distribution. The construction separates the finite-length design into two stages: the base matrix fixes the degree distribution and the first girth constraints, and a cyclic lift randomizes edge connections subject to exact algebraic checks. As a detailed example, we carry one \((3,10)\)-regular base through the lift and decoding stages. For this example, the selected 64-fold lift gives a code whose same-type Tanner graphs have girth at least eight, and it also excludes a specified weight-16 nondegenerate logical-support orbit. The resulting instance is a \([[10240,4108,\,10\le d\le32]]\) CSS code. For decoding, we use joint log-domain belief propagation together with low-complexity deterministic post-processing rules for small residual syndromes, including repairs for residual patterns with two unsatisfied checks. The frame error rate (FER) measurements provide finite-length decoding data for this detailed example; at depolarizing probability \(p=0.058\), the post-processing FER is \(1.0\times10^{-7}\).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper introduces a two-branch multiplicative-coset construction over finite fields for regular CSS LDPC base matrices. For target column weight J and even row weight L, it reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion to explicit quotient-coset conditions. Normalized exhaustive search yields bases for multiple (J,L) pairs; a detailed (3,10) example is lifted 64-fold to produce the [[10240,4108,10≤d≤32]] CSS code whose same-type Tanner graphs have girth ≥8 and exclude a specified weight-16 logical orbit, with joint BP decoding plus post-processing yielding FER=1.0×10^{-7} at p=0.058.

Significance. If the reduction and instance checks hold, the work supplies an algebraic, searchable method for regular CSS bases that is not tied to one degree distribution and cleanly separates base design from cyclic lifting. The concrete finite-length code instance together with decoding data adds to the limited set of explicit CSS LDPC performance results.

major comments (3)
  1. [detailed example section on the 64-fold lift] In the detailed example (the paragraph describing the 64-fold cyclic lift), the claims that the lift preserves girth ≥8 in the same-type Tanner graphs and excludes the weight-16 nondegenerate logical-support orbit rest solely on instance-specific algebraic checks; no general lemma is supplied showing that satisfaction of the base quotient-coset conditions plus these checks is sufficient to exclude all new short cycles or maintain CSS orthogonality for arbitrary lifts under the two-branch construction.
  2. [results paragraph on the lifted code] For the [[10240,4108]] instance, the distance interval 10≤d≤32 is stated without any derivation, computational method, or reference to a supporting theorem or search procedure that establishes the lower bound of 10.
  3. [decoding performance paragraph] The reported post-processing FER of 1.0×10^{-7} at depolarizing probability p=0.058 supplies no information on the number of simulated frames, statistical error bars, or the precise deterministic post-processing rules applied to residual syndromes with two unsatisfied checks.
minor comments (1)
  1. [construction and search paragraph] The phrase 'normalized exhaustive search' is introduced without a definition of the normalization criterion or the precise search procedure; a short clarifying sentence would aid reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments on our manuscript. We address each major comment below, indicating where revisions will be made to improve clarity and completeness.

read point-by-point responses

  1. Referee: In the detailed example (the paragraph describing the 64-fold cyclic lift), the claims that the lift preserves girth ≥8 in the same-type Tanner graphs and excludes the weight-16 nondegenerate logical-support orbit rest solely on instance-specific algebraic checks; no general lemma is supplied showing that satisfaction of the base quotient-coset conditions plus these checks is sufficient to exclude all new short cycles or maintain CSS orthogonality for arbitrary lifts under the two-branch construction.

    Authors: We agree that the girth and orbit-exclusion claims for the lifted code are established via instance-specific algebraic checks rather than a general lemma. The two-branch construction reduces the base-level conditions to quotient-coset requirements, after which the cyclic lift is selected and verified explicitly for the chosen parameters; we do not claim that the base conditions alone guarantee the properties for every possible lift. We will revise the detailed-example section to state this limitation explicitly and to clarify that the reported girth ≥8 and orbit exclusion apply to the specific 64-fold lift examined. revision: yes

  2. Referee: For the [[10240,4108]] instance, the distance interval 10≤d≤32 is stated without any derivation, computational method, or reference to a supporting theorem or search procedure that establishes the lower bound of 10.

    Authors: The lower bound d ≥ 10 is obtained from an exhaustive enumeration of low-weight logical operators consistent with the girth properties of the same-type Tanner graphs; the upper bound follows from the existence of a weight-32 logical operator. We will add a concise paragraph (or appendix reference) describing the computational procedure used to obtain these bounds. revision: yes

  3. Referee: The reported post-processing FER of 1.0×10^{-7} at depolarizing probability p=0.058 supplies no information on the number of simulated frames, statistical error bars, or the precise deterministic post-processing rules applied to residual syndromes with two unsatisfied checks.

    Authors: We acknowledge the omission of simulation statistics and the exact post-processing rules. We will expand the decoding-performance paragraph to report the total number of simulated frames, note the absence of error bars at the observed error rate, and provide a precise description of the deterministic post-processing rules, including the handling of residual syndromes with exactly two unsatisfied checks. revision: yes

Circularity Check

0 steps flagged

No circularity; algebraic reduction and instance verification are self-contained

full rationale

The paper defines a two-branch multiplicative-coset construction that maps regularity, CSS orthogonality, and 4-cycle exclusion to explicit quotient-coset conditions over a finite field, solved via normalized exhaustive search for base matrices. The 64-fold cyclic lift is then applied subject to algebraic checks that are performed for the specific (3,10) example. No step equates a derived claim to its own input by construction, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content reduces to the present work. The derivation is a constructive method with external algebraic verification and is therefore self-contained against the listed circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard finite-field arithmetic and the assumption that exhaustive search over cosets will locate valid bases; no free parameters or invented entities are introduced in the abstract description.

axioms (1)
  • domain assumption Finite fields admit multiplicative subgroups whose cosets can encode regularity, CSS orthogonality, and 4-cycle exclusion simultaneously.
    The construction treats this encoding as feasible and searchable for chosen J and L.

pith-pipeline@v0.9.0 · 5812 in / 1360 out tokens · 38655 ms · 2026-05-25T03:59:35.658423+00:00 · methodology

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Reference graph

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