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Cyclic Codes of Length 7_p^s over F_p^m + uF_p^m : Characterization, Duals, and Applications to Quantum and LCD Codes

Payel Chandra, Kalyan Hansda

arXiv:2609.07181v1cs.ITmath.IT

TL;DR

For cyclic codes of length 7p^s over R2, explicit classification, counting, and dual-code results had remained open. Under conditions making Φ7 irreducible, the paper decomposes the codes and classifies their cyclotomic component; it also constructs CSS quantum codes and characterizes LCD codes.

  • Problem

    An explicit classification, exact cardinality evaluation, and dual-code derivation for cyclic codes of length 7p^s over R2 remained open.

    Source: §1 · p. 1

  • Method

    Under conditions making Φ7 irreducible, the paper decomposes the codes and classifies cyclotomic codes using residue and torsion subcodes, then derives dual generators.

    Source: §2.1 · p. 2, §3.1 · p. 6, §3.3 · p. 9

  • Results

    The paper establishes four disjoint generator-based types, exact counting formulas, explicit Euclidean dual generators, and applications to QECC and LCD codes.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The classified ring codes are used to construct quantum stabilizer code families via CSS and to characterize linear codes with complementary duals.

    Source: Conclusion · p. 15

    Exact minimum distance is computationally prohibitive for general repeated-root ring-linear codes; lower bounds can be obtained with the BCH bound on component cyclic subcodes.

    Source: §4.1 · p. 14

Abstract

from arXiv · show

Let $R_2 = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ ($u^2 = 0$), where $p$ is an odd prime and $m, s \in \mathbb{N}$. For $p \equiv 3, 5 \pmod 7$ with $\gcd(m, 6) = 1$, the cyclotomic polynomial $\Phi_7(x)$ is irreducible over $\mathbb{F}_{p^m}$. This yields a direct sum decomposition $C = C_1 \oplus C_2$ for any cyclic code $C$ of length $7p^s$ over $R_2$, where $C_1$ has length $p^s$ and $C_2$ is a $7$-cyclotomic code of length $6p^s$. We classify $7$-cyclotomic codes into four disjoint generator-based types and calculate exact cardinalities using residue and torsion subcodes. Furthermore, explicit generators for the Euclidean dual codes $C^\perp$ are determined. As operational applications of these classified codes, we construct new families of quantum stabilizer codes via the CSS framework and establish parameter criteria for linear codes with complementary duals (LCD codes).

1. Introduction

The paper addresses the open classification, counting, and dual-code problems for cyclic codes of length 7p^s over R2. Under specified conditions, it decomposes these codes and develops classifications and applications.

1. Introduction

  • Repeated-root cyclic codes arise when p divides the length, making their structure more intricate through nilpotent elements and non-principal ideal chains.Cyclic codes over R2 correspond to ideals of R2[x]/⟨x^n − 1⟩.Source: §1 · p. 1
  • An explicit classification, exact cardinalities, and dual-code derivation for length-7p^s cyclic codes over R2 remained open despite prior work on other prime-multiple lengths.The cited prior classifications included lengths p^s through 5p^s and general ℓp^s.Source: §1 · p. 1
  • When p ≡ 3, 5 (mod 7) and gcd(m, 6) = 1, Φ7(x) is irreducible over F_{p^m}, enabling a unique split C = C1 ⊕ C2.C1 has length p^s and C2 is a 7-cyclotomic code of length 6p^s.Source: §1 · p. 2
  • The paper classifies 7-cyclotomic codes into four generator-based types, derives exact counting formulas, computes Euclidean dual generators, and applies the codes to QECCs and LCD codes.The counting uses residue and torsion code techniques derived from the rank-nullity theorem.Source: §1 · p. 2

2. Preliminaries

The preliminaries establish the ring and cyclotomic conditions used in the classification, recall length-p^s codes and the Gray map, and state the CSS and LCD framework.

2. Preliminaries

  • The ring R2 has nilpotent u with u^2 = 0 and maximal ideal ⟨u⟩; its residue field is F_{p^m}.Elements have the form x0 + ux1, and units are characterized by x0 ∈ F_{p^m}^×.Source: §2.1 · p. 2

2.1. Ring Structure and Cyclotomic Polynomials

  • Φ7(x) is irreducible in F_{p^m}[x] if and only if p ≡ 3 or 5 (mod 7), with m an odd integer such that gcd(m, 6) = 1.Source: §2.1 · p. 3

2.2. Repeated-Root Cyclic Codes of Length ps over R2

  • Prior work partitions length-p^s cyclic codes over R2 into four structural types, including generator forms with parameters β, i, a(x), and γ.The cited results also provide an index parameter and cardinality formulas, though their expressions are not included in these passages.Source: §2.2 · pp. 3–4

2.3. Gray Map

  • The Gray map Φ sends a + ub to (b, a + b) and maps a ring-linear code of size p^{mk0} to a field-linear code of length 2n and dimension k0.It also preserves Euclidean duality.Source: §2.3 · p. 4

2.4. Quantum Error-Correcting Codes and LCD Codes

  • The CSS construction turns a dual-containing ring-linear code into a quantum stabilizer code, while LCD codes are characterized by trivial intersection with their Euclidean dual.The Gray image preserves the LCD property.Source: §2.4 · pp. 4–5

3. Cyclic Codes over Fpm + uFpm

The section studies cyclic codes of length 7p^s by splitting them into length-p^s and length-6p^s components, classifying the 7-cyclotomic codes, and counting codewords via residue and torsion subcodes. It also derives dual-code formulas for the 7-cyclotomic and length-p^s components and gives self-orthogonality criteria for Type 4 codes.

3. Cyclic Codes over Fpm + uFpm

  • Every cyclic code C of length 7p^s decomposes uniquely as C1 ⊕ C2, with component lengths p^s and 6p^s, respectively.Here C1 is cyclic and C2 is 7-cyclotomic; the decomposition applies when p ≡ 3 or 5 (mod 7), m is odd, and gcd(m, 6) = 1.Source: §3 · p. 5

3.1. Classification of Cyclotomic Codes of Length 7ps over Fpm + uFpm

  • The 7-cyclotomic codes fall into four mutually disjoint types: trivial, pure torsion, principal mixed, and two-generator non-principal ideals.Their residue and torsion ideals are ⟨π^β⟩ and ⟨π^γ⟩ with γ ≤ β; the indices and unit status of a(π) distinguish the types and exhaust the ideals.Source: §3.1 · pp. 5–7
  • The torsion index V for a principal mixed generator is β when a(x) = 0, and min{β, p^s − β + i} when a(x) ≠ 0.This index determines the torsion ideal ⟨g(x)^V⟩ associated with the generator ⟨g(x)^β + ug(x)^i a(x)⟩.Source: §3.1 · p. 7, §3.2 · p. 7

3.2. Enumeration of Codewords via Residue and Torsion Code Techniques

  • For any linear code C over R2, |C| = |Res(C)| · |Tor(C)|; for C = C1 ⊕ C2, |C| = |C1| · |C2|.For matched Type-4 components, nC = p^(7m(2p^s − β − γ)); matched Type-2 components have nC = p^(7m(p^s − β)).Source: §3.2 · pp. 7–8

3.3. Dual Codes of 7-Cyclotomic and Cyclic Codes

  • Duals are derived using the cardinality identity |C| · |C⊥| = |R2[x]/⟨f(x)^p^s⟩| alongside annihilator containment.The cyclotomic factor g(x) is self-reciprocal, while g′(x) = x − 1 has reciprocal −g′(x).Source: §3.3 · p. 9

3.3.1. Dual Codes of 7-Cyclotomic Codes of Length 6ps

  • The trivial cyclotomic codes are dual to each other, and the dual of C2(β) = ⟨ug(x)^β⟩ is ⟨g(x)^(p^s−β), u⟩.For Type 3, the dual generator changes form according to whether a(x) vanishes and whether p^s is at least 2β − i.Source: §3.3.1 · pp. 9–10
  • For Type 3 with a(x) ≠ 0, the dual is C3(p^s − β, p^s + i − 2β + v, −G(x)) if p^s ≥ 2β − i, and C3(β − i, v, −G(x)) if p^s ≤ 2β − i.Source: §3.3.1 · p. 10
  • For Type 4 7-cyclotomic codes, the dual is C4(p^s−γ,0,0,p^s−β) when a(x)=0; otherwise it is one of two C3 codes, selected by whether p^s ≥ β+γ−i.For a(x)≠0 and p^s ≥ β+γ−i, the parameters are (p^s−γ, p^s+i−β−γ+t, −G(x)); for p^s ≤ β+γ−i, they are (β−i, t, −G(x)).Source: §3.3.1 · pp. 11–12
  • Under γ < min{β,p^s−β+i}, Type 4 is self-orthogonal for a(x)=0 iff β+γ≤p^s; for a(x)≠0 and p^s≥β+γ−i, iff β+γ≤p^s and p^s+i−β−2γ+t≥0.Source: §3.3.1 · p. 12
  • For length-p^s cyclic codes, reciprocation sends g′(x)=x−1 to −g′(x), producing alternating (−1)^k factors in dual-code unit parameters; Proposition 3.13 characterizes Types 2–4.Source: §3.3.1 · p. 12
  • For a(x)≠0 and p^s≥2β−i, the Type 3 length-p^s dual is C′3(p^s−β,p^s+i−2β+t,(−1)^(p^s+i+1)G(x)).Source: §3.3.1 · p. 12
  • For Type 4 length-p^s codes with a(x)≠0, the dual is C′3 with parameters (p^s−γ,p^s+i−β−γ+t,(−1)^(p^s+i−β−γ+1)G(x)) if p^s≥β+γ−i, and (β−i,t,−G(x)) if p^s≤β+γ−i.Source: §3.3.1 · p. 12

4. Applications: Quantum Error-Correcting and LCD Codes

The applications use the classified cyclic codes and their duals to construct CSS quantum stabilizer codes and to characterize LCD codes. The paper concludes with its classification, enumeration, and dual-code results, while noting possible extensions to larger primes and higher nilpotency.

4.1. Quantum CSS Construction Framework

  • Type 3 codes with a(x)=0 yield CSS codes when β≤⌊ps/2⌋, with parameters [[14ps, 14(ps−2β), dq]]pm.The construction uses matched component codes and the CSS framework.Source: §4.1 · p. 13
  • For Type 3 with a(x)≠0, CSS construction requires ps≥2β−i and ps+t≥2β, yielding [[14ps, 14(ps−2β), dq]]pm codes.The passage gives the parameter conditions for self-orthogonality.Source: §4.1 · p. 13
  • Type 4 codes with a(x)=0 yield [[14ps, 14(ps−β−γ), dq]]pm codes when β+γ≤ps.The construction uses matched two-generator component codes.Source: §4.1 · p. 13
  • In the other cases covered by Propositions 3.9 and 3.11, C⊥ is not contained in C, so CSS stabilizer codes cannot be constructed.Source: §4.1 · p. 13
  • The CSS distance dq is lower-bounded by the smaller of the residue- and torsion-code Hamming distances; exact general distances are computationally prohibitive to determine.The BCH bound can provide lower bounds from component cyclic subcodes over Fpm.Source: §4.1 · p. 14
  • The Gray map preserves the LCD property; the stated LCD cases include pure torsion Type 2 at β=0 and Type 3 with a(x)=0 at β=0 or β=ps.The pure-torsion case C=⟨u⟩ has a Gray image that is an [14ps, 7ps, dH]pm LCD code.Source: §4.1 · pp. 14–15

Conclusion

  • The conclusion reports a complete classification and enumeration, explicit Euclidean dual generators, and applications to quantum stabilizer and LCD codes.It suggests extending the framework to larger-prime constacyclic codes and chain rings with nilpotency q≥3.Source: Conclusion · p. 15

Declaration

  • The authors report using generative AI tools only for LaTeX typesetting and English grammar, with mathematical work independently performed by the authors.Source: Declaration · p. 15

Appendix A. Computational Verification and Exemplary Quantum Codes

The appendix reports computational verification and provides exemplary quantum-code parameters.

Appendix A. Computational Verification and Exemplary Quantum Codes

  • The reported cyclotomic factorizations, ideal annihilators, and code cardinalities were verified using Magma Computational Algebra System Version 2.26.Source: Appendix A · p. 18
  • Table A.1 summarizes exemplary quantum stabilizer-code parameters for choices of p, s, m, and structural parameters.Source: Appendix A · p. 18, Table A.1 · p. 18
  • The table is titled as a collection of new [[N, K, dq]]pm codes from 7ps-length cyclic codes over R2.Source: Table A.1 · p. 18
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