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Low-Weight Canonical Logical Bases from Pair-Partition Codes
Koki Okada, Kenta Kasai
TL;DR
Short cyclic shifts of seeds need not span the full logical space, leaving representative choice, canonical pairing, and completeness to be established together. The paper constructs cofactor-based canonical bases under an invertibility condition, with quaternary coefficients for CPM–PP checks. The construction yields complete logical pairs, while weight preservation requires an additional pure-shift condition.
Problem
Cyclic shifts of short seeds need not span the full logical space, and representative choice and X/Z pairing remain design tasks alongside check construction.
Source: §1 · p. 1Method
The paper assigns quaternary coefficients to CPM–PP checks and uses cofactor representatives with an inverse pairing polynomial to construct canonical logical bases.
Source: §1 · pp. 1–2Results
All seven instances have check-row weight 10 on both sides, and the construction proves the logical classes complete.
Source: §2.5 · p. 6, §3.5 · p. 11Takeaways & Limitations
The construction provides explicit bases that jointly establish kernel conditions, canonical pairing, and completeness.
Source: §1 · pp. 1–2Weight preservation is not guaranteed by invertibility alone: a multi-term inverse can combine shifted seeds and increase their weight.
Source: §2.3 · p. 5
Abstract
from arXiv · showhide
We construct complete canonical logical bases for qubit CSS codes by assigning quaternary coefficients to binary circulant permutation matrix pair-partition (CPM-PP) checks, constructing and normalizing logical representatives, and expanding the result into binary matrices. When the two check systems have invertible submatrices on disjoint column sets, cofactor representatives are canonically paired by the inverse of a single pairing polynomial. The resulting pairs span the entire logical space. When the pairing polynomial is a cyclic-shift monomial with coefficient one, normalization preserves the binary weights of the representatives. We state the construction for general block dimensions and CPM size, work through a corresponding example, and report the parameters, check ranks, and basis weights of seven binary codes. Representative examples have parameters $[[320,80,14]]$, $[[448,112,18]]$, and $[[2048,512,24]]$. All three have maximum check weight 10 on both the X and Z sides. Their canonical logical representatives have binary weights 25, 27, and 27, respectively, on both the X and Z sides.
1 Introduction
The paper constructs complete canonical logical bases that retain the cyclic structure of CPM–PP checks. It combines cofactor representatives and canonical pairing under an invertibility condition, using quaternary coefficients to meet that condition for fully populated binary CPM arrays.
1 Introduction
- The construction produces k logical representatives per side and pairs them so matching X/Z representatives anticommute while different-index pairs commute.The identities also ensure representatives commute with the checks and preserve the code space.Source: §1 · p. 1
- Cyclic shifts of short seeds need not span the full logical space, motivating a construction that establishes kernel conditions, canonical pairing, and completeness together.The framework in [4, Secs. II and VII] treats representative choice and X/Z pairing as design tasks alongside check construction.Source: §1 · p. 1
- When both check systems have invertible submatrices on disjoint column sets, cofactor seeds share a scalar polynomial pairing whose inverse normalizes all logical pairs.The invertible submatrices determine check ranks, allowing the number of pairs to be shown equal to the full logical dimension.Source: §1 · pp. 1–2
- Quaternary coefficients enable the invertibility condition for fully populated binary CPM arrays with J > 1, while preserving commutation.Evaluating binary monomial entries at z = 1 yields an all-one matrix, making every J × J minor non-unit; the assigned coefficients can remove this obstruction.Source: §1 · p. 2
- The construction is stated for general J, L, and P, then applied with J = 3, L = 8, P = 20 to report seven code instances and their canonical bases.In the worked example, coefficient assignment and companion-matrix expansion raise row weight from 8 to 10; weight preservation during canonicalization is guaranteed under the pure-shift condition on g.Source: §1 · p. 2
2 From CPM–PP checks to binary canonical logical bases
The section develops a four-stage construction of complete canonical logical bases for CPM–PP CSS codes, from commuting checks to binary representatives. Disjoint invertible check submatrices and an invertible pairing polynomial provide the conditions for canonical pairing and completeness.
2.1 Stage 1: Construct binary CPM–PP check matrices
- Paired CPM shifts cancel in characteristic two, establishing the commutation condition for the binary CPM–PP checks.The pair-partition cancellation does not require P to be prime.Source: §2.1 · pp. 2–3
2.2 Stage 2: Assign quaternary coefficients preserving PP cancellation
- Quaternary coefficients assigned equally to each paired shift preserve the cancellation and ensure CSS commutation; invertibility is checked separately.The coefficients replace the nonzero entries of the CPMs.Source: §2.2 · p. 3
2.3 Stage 3: Construct canonical logical bases in three steps
- Cofactor seeds built from disjoint column sets satisfy the check-kernel conditions, and each seed is confined to J + 1 block columns.The free-column entries are adjusted using determinants of the selected check submatrices, yielding sparse seeds.Source: §2.3 · p. 4
- A unit pairing polynomial normalizes the seed pairs; its inverse gives canonical pairings, and g = z^ν preserves binary weight while producing P cyclic shifts per pair.All seven reported instances satisfy the stronger condition g = z^ν with coefficient one.Source: §2.3 · p. 5
2.4 Stage 4: Expand checks and logical bases into binary matrices
- Companion-matrix expansion carries the required products and transposes into binary matrices, where each check has rank 2JP and the code has n = 2LP and k = 2(L − 2J)P.These rank and parameter conclusions hold under the unit condition in Stage 3.Source: §2.4 · p. 6
2.5 Proof outline: Validity and completeness
- The sP canonical pairs are independent and match the logical-space dimension, so they form complete bases; the disjoint invertible-submatrix condition must be checked for each construction.The condition need not hold for every PP code.Source: §2.5 · pp. 6–7
3 A worked construction and seven code instances
The worked example applies the four construction stages at J = 3, L = 8, and P = 20; the same procedure yields seven binary codes with complete canonical logical bases.
3.1 Stage 1 example: Binary CPM–PP checks
- The example's exponent arrays make paired CPM shifts agree, satisfying the pair-partition condition used to construct the binary checks.For one pair, the passage gives d0,1 − e0,1 = 9 = d0,7 − e0,7.Source: §3.1 · p. 7
3.2 Stage 2 example: Quaternary coefficients and polynomial checks
- Quaternary coefficients satisfy the coefficient condition in the example, and substitution gives 3 × 8 polynomial check matrices over R20.For the pair {1, 7}, the products are ωω2 = 1 and ω2ω = 1.Source: §3.2 · p. 8
3.3 Stage 3 example: Constructing and canonicalizing logical representatives
- With free columns 3 and 6, the example constructs 40 candidate representatives per side and normalizes their pairing using g = z6 and g−1 = z14.The pairing shifts partners by six within each group of 20; this pure shift changes positions without changing coefficients, so normalization does not increase weight.Source: §3.3 · p. 9
3.4 Stage 4 example: Binary companion-matrix expansion
- Binary companion-matrix expansion produces 120 × 320 check matrices and 80 × 320 logical matrices; each logical-basis row has binary weight 27.The expanded checks have row weight 10 on both sides, while the binary check ranks are both 120 and the logical dimension is 80.Source: §3.3 · p. 10, §3.4 · p. 10
3.5 Seven codes and their complete canonical logical bases
- All seven instances have check-row weight 10 on both sides, and their 4P representatives per side cover all logical degrees of freedom.For the [[2048, 512, 24]] instance, the reported distance is exact.Source: §3.5 · pp. 11–12, Table 2 · p. 12
- The [[2048, 512, 24]] instance has exact distance 24, certified by excluding nontrivial logical operators of weight at most 23 and checking an operator of weight 24.The reported distance search uses binary costs for quaternary symbols and tests nontriviality by pairing with the complete opposite logical basis.Source: §3.5 · p. 12
Declaration of generative AI use
The authors used OpenAI Codex to assist with manuscript drafting and revision, mathematical work, and computational verification; they reviewed the AI-assisted material and take responsibility for the final content.
Declaration of generative AI use
- The authors report using OpenAI Codex for drafting, translation, revision, mathematical arguments, computational verification code, and examination of numerical results.They state that they reviewed the AI-assisted material and take full responsibility for the manuscript's mathematical claims, computational results, references, and final content.Source: Declaration of generative AI use · p. 12