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Quantum MDS codes from complements of unions of finite-field subsets

Naihong Hu, Hong Ji

arXiv:2609.09943v1cs.ITmath-phmath.ITmath.MPmath.QA

TL;DR

Quantum MDS codes with new lengths and minimum distances greater than q/2 + 1 remain an important construction problem. This paper uses complements of finite-field subset unions to construct five Hermitian self-orthogonal GRS families and corresponding quantum MDS families, with distance and comparison gains under stated conditions.

  • Problem

    For quantum MDS codes longer than q + 1, attainable parameters depend more strongly on the construction, and new lengths with minimum distance greater than q/2 + 1 remain an important problem.

    Source: §1 · p. 2

  • Method

    The paper uses complements of unions of finite-field subsets as GRS locators and applies vanishing-polynomial and Frobenius conditions to construct five Hermitian self-orthogonal families.

    Source: §1 · p. 3, §5 · p. 31

  • Results

    For fixed parameter-set counts, the families have lengths q^2 − O(q) and can attain minimum distances q − O(1), with d* > q/2 + 1 for sufficiently large q.

    Source: §1 · p. 4

  • Takeaways & Limitations

    Allowing selected subsets to intersect leaves the dimension bound independent of intersection sizes, while the code length depends on the union cardinality.

    Source: §1 · p. 3

    The first-family parameter choice assumes a + b ≤ q − 1, with the selected Ni and Di sizes bounded as specified.

    Source: §3.3 · p. 13

Abstract

from arXiv · show

Let $q$ be an odd prime power. We use complements of unions of subsets of $\mathbb F_{q^2}$ as locator sets and establish a sufficient condition under which a generalized Reed--Solomon (GRS) code is Hermitian self-orthogonal. Using cosets of multiplicative subgroups and sets with prescribed trace or norm values, we construct five families of Hermitian self-orthogonal GRS codes over $\mathbb F_{q^2}$. The Hermitian construction then yields five corresponding families of $q$-ary quantum maximum-distance-separable (MDS) codes. Under suitable parameter conditions, these quantum codes have minimum distances greater than $q/2+1$. By comparing codes of the same length, we give conditions under which our codes have strictly larger minimum distances than those obtainable from several previously known constructions based on trace maps, linear transformations, and cosets of multiplicative subgroups, either directly or via the propagation rule. We further show that such improvements occur for infinitely many values of $q$.

1. Introduction

The paper constructs Hermitian self-orthogonal GRS codes from complements of unions of finite-field subsets, then uses them to obtain quantum MDS families with new parameter ranges. It compares their minimum distances against prior constructions at equal lengths.

1. Introduction

  • For quantum MDS lengths greater than q + 1, attainable parameters depend more strongly on the construction, and new lengths with d > q/2 + 1 remain an important problem.Earlier trace-subspace and trace-norm constructions record intersection sizes in their length formulas.Source: §1 · pp. 2–3
  • The construction takes locators as complements of unions and uses vanishing polynomials and Frobenius behavior to choose column-multiplier norms, allowing the selected subsets to intersect.The dimension bound does not explicitly depend on intersection sizes, although the length depends on the union cardinality.Source: §1 · p. 3
  • The five GRS families combine prescribed norm values, prescribed trace values, and multiplicative-subgroup cosets in different configurations.The families include norm-value/coset, trace-value/coset, trace/norm, and two mixed trace-value constructions.Source: §1 · p. 3
  • The five quantum MDS families have length n = q^2 − Sq + θ and exist for 2 ≤ d ≤ q − S + 1; for fixed set counts, d can exceed q/2 + 1 for sufficiently large q.Their lengths are q^2 − O(q), and their minimum distances can reach q − O(1).Source: §1 · pp. 3–4
  • At equal lengths, the paper gives conditions for strictly larger minimum distances than trace-based and linear-transformation constructions, and reports such gains for infinite subfamilies.For multiplicative-subgroup coset constructions, it derives fixed-length upper bounds for R5–R16 and identifies R17–R20 as too short to reach the lengths covered by a stated comparison.Source: §1 · p. 4

2. Preliminaries

The preliminaries establish the finite-field setting, define GRS locators and multipliers, and connect Hermitian self-orthogonality to dual-code criteria.

2. Preliminaries

  • Throughout, q is an odd prime power, and the trace and norm maps are defined on the extension Fq2/Fq.The paper writes Tr(x) = x + x^q and N(x) = x^(q+1).Source: §2 · p. 5
  • A GRS code evaluates polynomials at distinct locators and scales the evaluations by nonzero column multipliers; it is an MDS code with parameters [n, k, n − k + 1]q2.The locators are the evaluation points, and each multiplier is indexed by its corresponding locator.Source: §2 · p. 5
  • The Hermitian self-orthogonality criterion uses a polynomial degree bound and a multiplier relation to ensure the GRS code lies in its Hermitian dual.Hermitian dual membership is related to Euclidean dual membership through q-conjugation.Source: §2 · pp. 5–6

3. Hermitian self-orthogonal GRS codes

Section 3 develops a complement-of-unions criterion for Hermitian self-orthogonal GRS codes and supplies intersection and vanishing-polynomial formulas for its finite-field subsets.

3. Hermitian self-orthogonal GRS codes

  • The section constructs Hermitian self-orthogonal GRS codes using locator sets formed as complements of unions in Fq2 or Fq2*.Its applications use multiplicative-subgroup cosets and sets with prescribed trace or norm values.Source: §3 · p. 6

3.1. A general construction

  • For A = X \ B, the Lagrange coefficients on A can be expressed through the vanishing polynomial of B, including when B is a union of intersecting subsets.Theorem 3.3 gives a sufficient condition using a Frobenius relation and a degree bound to produce a Hermitian self-orthogonal MDS GRS code.Source: §3.1 · pp. 7–8
  • A specialization handles unions of multiplicative-subgroup cosets by choosing a polynomial factor whose Frobenius behavior meets the general criterion.Lemma 3.4 supplies a further sufficient condition for a Hermitian self-orthogonal MDS GRS code from this setup.Source: §3.1 · pp. 8–9

3.2. Intersection formulas and vanishing polynomials

  • The norm-level cosets Uν partition Fq2* into sets of size q + 1, while the sets Mδ partition Fq2* into sets of size q − 1.These partition and intersection formulas support the code-length calculations and polynomial choices used in the constructions.Source: §3.2 · p. 9
  • The trace-coordinate formulas give Tr(s + tω) = 2s, Tr(−ωx) = −2Δt, and N(s + tω) = s^2 − Δt^2.The sets defined by either trace condition have cardinality q.Source: §3.2 · pp. 10–11

Their intersection consists of the single point

The section derives intersection and vanishing-polynomial tools for trace-, norm-, and multiplicative-coset subsets, then uses them to construct five families of Hermitian self-orthogonal GRS MDS codes.

Their intersection consists of the single point

  • The intersection formulas and monic vanishing polynomials for the selected trace, norm, and multiplicative-coset sets support the five subsequent constructions.The vanishing polynomials have only simple roots.Source: Their intersection consists of the… · pp. 12–13

3.3. The first family of Hermitian self-orthogonal GRS codes

  • For 1 ≤ k ≤ q−a−b, prescribed-norm sets and multiplicative-subgroup cosets yield Hermitian self-orthogonal [N1, k, N1−k+1]q2 MDS codes.The construction assumes a+b ≤ q−1.Source: §3.3 · p. 13

3.4. The second family of Hermitian self-orthogonal GRS codes

  • Replacing the norm-value sets with prescribed-trace sets gives Hermitian self-orthogonal [N2, k, N2−k+1]q2 MDS codes for 1 ≤ k ≤ q−u−b−1.The construction assumes u+b ≤ q−2.Source: §3.4 · p. 14

3.5. The third family of Hermitian self-orthogonal GRS codes

  • Combining prescribed-trace and prescribed-norm sets gives Hermitian self-orthogonal [N3, k, N3−k+1]q2 MDS codes for 1 ≤ k ≤ q−R−a.The construction assumes R+a ≤ q−1.Source: §3.5 · p. 15

3.6. The fourth family of Hermitian self-orthogonal GRS codes

  • Combining two prescribed-trace set families with selected multiplicative cosets gives Hermitian self-orthogonal [N4, k, N4−k+1]q2 MDS codes for 1 ≤ k ≤ q−u−v−b−1.The construction assumes u+v+b ≤ q−2.Source: §3.6 · p. 16

3.7. The fifth family of Hermitian self-orthogonal GRS codes

  • Combining two prescribed-trace set families with prescribed-norm sets gives Hermitian self-orthogonal [N5, k, N5−k+1]q2 MDS codes for 1 ≤ k ≤ q−R−V−a.The construction assumes R+V+a ≤ q−1.Source: §3.7 · p. 17

4. Quantum MDS codes and parameter comparisons

The section uses five families of Hermitian self-orthogonal GRS codes to construct quantum MDS codes and compares their attainable minimum distances at fixed lengths with those of earlier constructions, including codes obtained by propagation. It derives distance bounds from propagation-rule length congruences and identifies strict improvements, including for infinitely many values of q.

4.1. Parameters of the five families of quantum MDS codes

  • For n = q^2 - Sq + θ, the five families give quantum MDS codes [[n, n - 2d + 2, d]]_q for every 2 ≤ d ≤ d* = q - S + 1.When S ≤ (q - 1)/2, the families include codes with minimum distance greater than q/2 + 1.Source: §4.1 · pp. 18–19

4.2. Comparison with constructions based on trace maps

  • For S ≥ 2 under condition (9), our codes strictly exceed the same-length distance upper bounds for both trace-based families, directly or via propagation.With selected trace values, norm values, and cosets fixed, this strict improvement occurs for all sufficiently large q.Source: §4.2 · p. 21

4.3. Comparison with constructions based on linear transformations

  • For target lengths above (q^2 - 1)/2, Proposition 4.5 determines the largest distance attainable by the first two linear-transformation families, including propagated codes.Corollary 4.6 gives the distance by cases in S and θ; if its value is below 2, those constructions yield no code of that length with distance at least 2.Source: §4.3 · pp. 22–23
  • At n = (q - R)(q - b), the second construction attains d* = q - S + 1 > q/2 + 1 and exceeds the linear-transformation benchmark by q - S + ⌈T/2⌉ - t > 0.Here S = R + b, T = S - h, and t = Rb - hq; under condition (12), the gain grows linearly with q for fixed R and b.Source: §4.3 · pp. 24–25
  • Each of five specified subfamilies has minimum distance greater than q/2 + 1 and strictly exceeds the same-length linear-transformation constructions, directly or via propagation.The subfamilies are given for q-ranges in Table 2; every row satisfies d* > dLLJ.Source: §4.3 · pp. 25–27

4.4. Comparison with constructions based on cosets of multiplicative subgroups

  • Propagation reduces length and minimum distance equally, while congruence restrictions on source lengths yield upper bounds for R5–R16 codes at a target length.For n=q^2−Sq+θ and d*=q−S+1, Corollary 4.10(i) applies when θ>S and q>2θ+4S+1.Source: §4.4 · pp. 27–29
  • No code from R17–R20, directly or after propagation, can attain the constructed length n because those families’ lengths are at most (q^2−1)/2.The propagation rule does not increase length.Source: §4.4 · pp. 29–31
  • With R,b≥2, S=R+b≥5, and q>2Rb+4S+1, n=(q−R)(q−b) supports quantum MDS distances through d*=q−S+1>q/2+1, strictly above R5–R16 at n.At the same length, the minimum-distance gain is at least q/2 − (Rb + 2S + 1/2).Source: §4.4 · p. 30
  • For fixed admissible R and b, the strict same-length distance improvement holds for infinitely many q, and its lower bound grows linearly with q.The gain is bounded below by q/2 − (Rb + 2S + 1/2) when S and Rb are fixed.Source: §4.4 · p. 31

5. Conclusion

The paper constructs five families of quantum MDS codes from Hermitian self-orthogonal GRS codes and compares their minimum distances with previously known constructions at equal lengths. It establishes conditions for strict improvements, including gains that grow linearly with q for fixed admissible parameters, and identifies extensions for future work.

5. Conclusion

  • The five quantum MDS families share n = q^2 − Sq + θ and 2 ≤ d ≤ q − S + 1, with θ depending on set cardinalities and intersection numbers.The last two families allow nonempty triple intersections.Source: §5 · p. 31
  • The paper gives sufficient conditions for strictly larger minimum distances than two trace-based constructions and determines the largest attainable distances for linear-transformation constructions in the considered range.Its common parameterization supports comparisons at equal lengths, including codes obtained by propagation.Source: §5 · p. 31
  • For coset-based constructions R5–R16, length congruences yield upper bounds on distances for original and propagated codes, while R17–R20 cannot reach the lengths covered by Corollary 4.11.The two-parameter family from the second construction satisfies one strict-improvement condition in that corollary's range.Source: §5 · pp. 31–32
  • For fixed admissible R and b, the second construction strictly improves on linear-transformation constructions, including propagated codes, for sufficiently large q, with a minimum-distance gain growing linearly with q.The comparison applies under the conditions in Corollary 4.11.Source: §5 · pp. 31–32
  • Extending the method to extended GRS codes and other structured finite-field subsets with explicitly computable intersections is identified as future work.Source: §5 · p. 32

Declaration of competing interest

The authors report no known competing financial interests or personal relationships that could have influenced the work. The accompanying appendix material summarizes selected quantum MDS constructions and includes two length forms from.

Declaration of competing interest

  • The authors declare no known competing financial interests or personal relationships that could have influenced the reported work.Source: Declaration of competing interest · p. 32
  • The appendix tables summarize selected recent and earlier quantum MDS constructions, with parameters satisfying the cited theorems' remaining hypotheses.Source: Declaration of competing interest · p. 33
  • Two listed constructions from have lengths (s + t)q − st and 1 + sq + t(q + 1) − N_{s,t}, respectively.The displayed parameter fragments give 1 ≤ s,t ≤ q for the first and 1 ≤ s,t ≤ q − 1 for the second.Source: Declaration of competing interest · p. 33

Appendix A. Selected parameters of previously known quantum MDS codes

Appendix A presents selected earlier quantum MDS families and a continuation of a table of selected recent families. Its listed parameter fragments include divisibility and range conditions, with integer distances d ≥ 2.

Appendix A. Selected parameters of previously known quantum MDS codes

  • The listed distance parameter d is an integer with d ≥ 2.Source: Appendix A · p. 33
  • Several listed entries constrain parameters through divisibility conditions involving h, λ, τ, or ρ, alongside bounds on r or σ.Source: Appendix A · p. 33
  • Appendix A includes selected earlier families and a continuation of the table of selected recent quantum MDS families.Source: Table A.2 · p. 34, Table A.1 · p. 34
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