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New distance bounds for one-generator quasi-cyclic codes with applications to Hermitian LCD codes and entanglement-assisted quantum codes

Kanat Abdukhalikov, Duy Ho, Rasha M. Shat

arXiv:2609.31300v1cs.ITmath.IT

TL;DR

The paper develops block-level distance bounds for one-generator quasi-cyclic codes and uses them in a search for Hermitian LCD codes. Six new codes improve corresponding Araya–Harada lower bounds, and the constructions yield maximal-entanglement EAQECCs with reported improvements and unrecorded parameter sets.

  • Problem

    The paper seeks stronger minimum-distance lower bounds for one-generator quasi-cyclic codes and good-parameter Hermitian LCD codes.

    Source: §1 · p. 2

  • Method

    Block-puncturing and block-shortening at generator-block level yield distance bounds for one-generator quasi-cyclic codes, including a nondecreasing chain for balanced codes beginning with the Jensen bound.

    Source: §1 · p. 2, §6 · p. 24

  • Results

    Six new Hermitian LCD codes improve the corresponding lower bounds in the Araya-Harada table.

    Source: §6 · p. 25

  • Takeaways & Limitations

    The Hermitian LCD codes yield 13 distinct EAQECCs that improve 24 entries in [31], plus 66 maximal-entanglement EAQECC parameter sets not recorded in Grassl’s table [19].

    Source: §6 · p. 25

    The stated setting assumes gcd(q, m) = 1, making x^m − 1 square free.

    Source: §3.2 · p. 10

Abstract

from arXiv · show

We introduce lower bounds on the minimum distance of one-generator quasi-cyclic codes of arbitrary index, obtained by puncturing and shortening the code at the level of its blocks. For balanced codes, these bounds form a nondecreasing chain whose first term is the Jensen bound. We apply these bounds to conduct a computer search and obtain many Hermitian LCD codes with good parameters. Among them there are six codes of length at most 30 which improve the corresponding lower bounds of Araya and Harada. These codes further give maximal-entanglement entanglement-assisted quantum codes, which we compare with the parameters currently recorded in the literature.

1 Introduction

The paper develops block-level distance bounds for one-generator quasi-cyclic codes, extending generator-based bounds to arbitrary index. It applies the bounds to search for Hermitian LCD codes and derives maximal-entanglement quantum codes from them.

1 Introduction

  • Quasi-cyclic codes have compact polynomial descriptions that support exhaustive searches while retaining algebraic structure for nontrivial minimum-distance bounds.They remain a source of record-holding linear and quantum codes.Source: §1 · p. 2
  • The algebraic approach treats quasi-cyclic codes as modules over R and uses CRT constituents or Jensen’s concatenated description to study them.The paper contrasts these methods with graph-theoretic analysis of quasi-cyclic LDPC codes.Source: §1 · p. 2
  • Existing algebraic distance bounds mostly use CRT or concatenated descriptions, while generator-based bounds included a condition specific to balanced codes of even index.The paper builds on the generator-based argument of Guan, Li, Liu and Ma [21].Source: §1 · p. 2
  • The new block-puncturing and block-shortening bounds apply to every one-generator quasi-cyclic code; for balanced codes, they form a nondecreasing chain beginning with the Jensen bound.The paper uses the bounds and a Hermitian LCD criterion in a computer search, reporting six quaternary codes of length at most 30 that improve Araya–Harada lower bounds [10, Table 6].Source: §1 · pp. 2–3
  • Hermitian LCD codes yield maximal-entanglement EAQECCs with parameters [[n, k, d; n−k]]_q, which the paper compares with Grassl’s table [19] and constructions by Li, Ezerman, Li, Ling and Sun [31].All computations used Magma.Source: §1 · p. 3

2 Preliminaries

The preliminaries represent quasi-cyclic codes as modules over a polynomial quotient ring, then describe their CRT constituents and concatenated structure. The latter supplies the Jensen minimum-distance bound.

2.1 Background on linear and quasi-cyclic codes

  • A quasi-cyclic code of length mℓ and index ℓ is invariant under the shift by ℓ coordinates and corresponds to a linear code over R of length ℓ.Here R = F[x]/⟨x^m − 1⟩, and the codewords can be represented by ℓ polynomials in R.Source: §2.1 · p. 4

2.2 The Chinese Remainder Theorem decomposition of quasicyclic codes

  • Assuming gcd(q, m) = 1, the Chinese Remainder Theorem decomposes the module into constituent linear codes of length ℓ over extension fields of F.The factorization of x^m − 1 is square-free under this assumption.Source: §2.2 · pp. 4–5

2.3 Concatenated decomposition and the Jensen bound

  • The concatenated structure pairs each constituent Ci with ⟨θi⟩ via ψi; its outer codes are the same as the constituents, and it yields the Jensen minimum distance bound for quasi-cyclic codes.Source: §2.3 · pp. 5–6

3 Block-support distance bounds for one-generator quasi-cyclic codes

Section 3 derives one-generator quasi-cyclic-code distance bounds from block puncturing and shortening. For balanced codes, the bounds form a nondecreasing chain beginning with the Jensen bound.

3.1 Block-support bound from block-shortened and block-punctured codes

  • Block puncturing and shortening yield a block-support lower bound for one-generator quasi-cyclic codes of arbitrary index.The puncturing operation deletes blocks outside a chosen subset; shortening restricts to codewords with a zero block before deleting that block.Source: §3.1 · pp. 7–9

3.2 Bounds for balanced one-generator quasi-cyclic codes

  • For balanced codes, the successive block-puncturing bounds are nondecreasing, and the first equals the Jensen bound.Balanced means that the generator components have the same gcd with x^m − 1; in this case, each block-shortened code is zero.Source: §3.2 · pp. 10–11, §3.3 · p. 14

3.3 The Jensen bound for balanced codes

  • In the [20, 4, 13]₄ example, the bounds rise from 8 to 10 to 12, while the actual distance is 13.The Jensen bound is 8; the two- and three-block puncturing bounds improve it to 10 and 12.Source: §3.3 · p. 14

4 Hermitian LCD one-generator quasi-cyclic codes

The section develops polynomial criteria for identifying Hermitian LCD one-generator quasi-cyclic codes and uses block-support distance bounds to screen a computer search. The search finds codes improving or matching known parameters, while balanced and unbalanced examples illustrate the bounds’ use.

4 Hermitian LCD one-generator quasi-cyclic codes

  • The search uses a polynomial Hermitian-hull description and LCD criterion to identify candidates, then applies distance bounds to screen them before exact minimum distances are computed.The paper also records balanced and unbalanced examples to illustrate the block-support bounds’ effectiveness in the search.Source: §4 · p. 15

4.1 Hermitian hulls and the LCD criterion

  • The Hermitian hull is the subcode Cλ; for a Hermitian LCD code, λ = h and this subcode is zero.Source: §4.1 · p. 18

4.2 Constructions and comparison with known parameters

  • Six searched codes improve the corresponding Araya–Harada lower bounds, and the search covers lengths up to 63.The six improvements are recorded in Table 3.Source: §4.2 · p. 19
  • Thirteen codes attain values listed in [10, Table 6], including [21]H4, which establishes dH4(21, 7) = 11.The search also gives an alternative one-generator quasi-cyclic construction of H4.Source: §4.2 · p. 19
  • For lengths n > 30, eleven codes match Grassl’s table [19], and five listed codes attain the best-known linear-code distance for their length and dimension.The code [4]H4 is reported as optimal.Source: §4.2 · p. 19

4.3 Performance of the block-support bounds

  • For balanced examples, Ψ(C) is infinite and the Jensen bound equals Φ1(C); for unbalanced examples, both Φ(C) and Ψ(C) are considered.The tables summarize the resulting bounds for the two cases.Source: §4.3 · p. 19, Table 4 · p. 20, Table 5 · p. 20

5 Quantum codes

Hermitian LCD codes yield maximal-entanglement EAQECCs with parameters [[n, k, d; n−k]]2. The paper compares these codes with recent constructions and Grassl’s table, reporting improvements and new parameter sets alongside a distance–entanglement trade-off distinction.

5 Quantum codes

  • Each [n, k, d]H4 Hermitian LCD code yields a maximal-entanglement EAQECC with parameters [[n, k, d; n−k]]2.The comparison considers rate, net rate, and error-correcting capacity.Source: §5 · pp. 20–21

5.1 Comparison with [31]

  • Thirteen distinct constructed EAQECCs strictly improve 24 entries in [31] under the paper’s comparison criterion.The 24 table rows count improved entries, not distinct constructed codes.Source: §5.1 · p. 21

5.2 Comparison with Grassl’s table [19]

  • The constructed codes yield 66 maximal-entanglement EAQECC parameter sets absent from Grassl’s table [19], and one improves its corresponding entry.The other 65 are characterized primarily as new parameter sets rather than improvements over.Source: §5.2 · p. 22
  • A positive ΔdL together with a positive Δc is a distance–entanglement trade-off, not an improvement under Definition 5.1.ΔdL measures increase over the table’s lower distance endpoint, while Δc measures change in required ebits.Source: §5.2 · p. 22

6 Conclusion

The paper introduces block-support distance bounds and applies them to a search for quaternary Hermitian LCD one-generator quasi-cyclic codes, then derives maximal-entanglement EAQECCs from those codes.

6 Conclusion

  • Block-support lower bounds apply to balanced and unbalanced one-generator quasi-cyclic codes of arbitrary index; for balanced codes, they form a nondecreasing chain whose first term recovers the Jensen bound.Source: §6 · p. 24
  • The paper gives a polynomial description of the Hermitian hull and a necessary and sufficient Hermitian LCD criterion, then uses them with the bounds to search quaternary codes up to length 63.Source: §6 · p. 24
  • Six new Hermitian LCD codes improve the corresponding lower bounds in the Araya-Harada table.Source: §6 · p. 25
  • The search produced thirteen one-generator quasi-cyclic Hermitian LCD codes attaining exact values recorded in the Araya-Harada table.Source: §6 · p. 25
  • The search also found the optimal Hermitian LCD code [4]H_4 and ten further codes of length greater than 30 matching best-known unrestricted quaternary linear-code distances at the same length and dimension.Source: §6 · p. 25
  • The resulting codes improve 24 entries in [31] with 13 distinct EAQECCs, and yield 66 maximal-entanglement EAQECCs absent from Grassl’s table [19].Source: §6 · p. 25
  • The authors identify extending block-support bounds to multigenerator quasi-cyclic and one-generator quasi-twisted codes, and systematic comparison with other bounds, as future work.Source: §6 · p. 25

Appendix

The appendix records generator polynomials for searched quaternary Hermitian LCD one-generator quasi-cyclic codes, organized by index, and explains the comparison labels used in the tables.

Appendix

  • Appendix tables record generator polynomials for searched quaternary Hermitian LCD one-generator quasi-cyclic codes, organized by index; captions show indices 2 (including a continuation), 3, 4, and 5.Source: Appendix · p. 29, Table A1 · p. 30, Table A2 · p. 31
  • For a code of length n = mℓ and index ℓ, the tables use F_4 = {0, 1, ω, ω^2}, with generator polynomials given in factorized form.Source: Appendix · p. 29
  • The Remark column compares minimum distance with unrestricted quaternary linear codes of the same length and dimension; OpLC means optimal, while BKLC means best-known and may not be optimal.Source: Appendix · p. 29
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