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Application of Constacyclic codes to Quantum MDS Codes
Bocong Chen, San Ling, Guanghui Zhang
TL;DR
The paper addresses the difficulty of constructing long quantum MDS codes with relatively large minimum distance by studying dual-containing constacyclic codes. It establishes existence conditions, constructs several dual-containing MDS constacyclic classes, and derives new quantum MDS codes with parameters exceeding previously available ones. The analysis is conducted under stated arithmetic and length assumptions.
Problem
Constructing q-ary quantum MDS codes with length n > q + 1 and relatively large minimum distance is challenging, while most known codes have distance at most q outside sparse lengths.
Method
The paper studies Hermitian dual-containing constacyclic codes, derives existence conditions, and uses them to construct MDS codes over Fq^2 for quantum-code construction.
Results
Four classes of q-ary quantum MDS codes are constructed, and many have minimum distances greater than those of available literature codes for fixed length and q.
Takeaways & Limitations
The results provide arithmetic criteria and constacyclic constructions for obtaining new quantum MDS codes with comparatively large minimum distance.
Takeaways & Limitations
The analysis assumes n is relatively prime to q and includes families restricted to specific arithmetic conditions such as q odd with 3 | (q + 1) and n = q^2 − 1.
Abstract
from arXiv · showhide
Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. To get $q$-ary quantum MDS codes, it suffices to find linear MDS codes $C$ over $\mathbb{F}_{q^2}$ satisfying $C^{\perp_H}\subseteq C$ by the Hermitian construction and the quantum Singleton bound. If $C^{\perp_{H}}\subseteq C$, we say that $C$ is a dual-containing code. Many new quantum MDS codes with relatively large minimum distance have been produced by constructing dual-containing constacyclic MDS codes (see \cite{Guardia11}, \cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on the existence condition for nontrivial dual-containing constacyclic codes. This would help us to avoid unnecessary attempts and provide effective ideas in order to construct dual-containing codes. Several classes of dual-containing MDS constacyclic codes are constructed and their parameters are computed. Consequently, new quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.
1 Introduction
The paper studies dual-containing constacyclic codes as a route to quantum MDS codes, deriving existence conditions and several new code classes. The resulting quantum MDS codes include parameters exceeding those previously available, including minimum distances above q in specified regimes.
- Problem setting: Quantum MDS codes are difficult to construct at lengths n > q + 1 and with relatively large minimum distance.Except for some sparse lengths such as n = q^2 + 1 and q^2 + 1, almost all known codes have minimum distance at most q.
- Motivation: Quantum MDS construction reduces to finding linear MDS codes C over Fq^2 satisfying C^⊥H ⊆ C.The Hermitian construction then yields q-ary quantum codes with parameters [[n, 2k − n, ≥d]]q.
- Research goal: The paper carefully studies when nontrivial dual-containing constacyclic codes exist.It derives elementary number-theoretic existence conditions to guide code construction and avoid unnecessary attempts.
- Results: Four classes of q-ary quantum MDS codes are constructed from dual-containing MDS constacyclic codes.The reported classes include cases with divisibility conditions 3 | (q + 1), 5 | (q + 1), and 7 | (q + 1), alongside a family where q = 10m + 3 or q = 10m + 7.
- Results: Many new codes have minimum distance greater than codes available in the literature for fixed length and q.The paper specifically reports minimum distance d > q^2 + 1 for class (i) when q > 11, and d > q for classes (ii) and (iii) under stated thresholds.
2 Existence conditions for nontrivial Hermitian dual-containing constacyclic codes
The section develops criteria for when nontrivial Hermitian dual-containing constacyclic codes exist, using generator polynomials and q^2-cyclotomic cosets. It then gives arithmetic conditions and examples establishing both existence and nonexistence in specific parameter settings.
- Constacyclic-code framework: A λ-constacyclic code is an ideal of Fq2[X]/⟨X^n−λ⟩ generated by a unique monic divisor of X^n−λ.Its defining set is a union of q^2-cyclotomic cosets, and the dimension is n−|Z|.
- Constacyclic-code framework: The BCH bound lower-bounds the minimum distance by d when the generator polynomial contains d consecutive specified roots.The Singleton bound gives k≤n−d+1 for an [n,k,d] linear code.
- Basic existence restrictions: A nontrivial dual-containing constacyclic code requires the constacyclic parameter to satisfy λ=λ^q, equivalently r|(q+1) when λ has order r.The inclusion result for nontrivial α- and β-constacyclic codes forces α=β, which supports the parameter restriction used here.
- Basic existence restrictions: For defining set Z, dual containment holds exactly when Z∩(−qZ)=∅ modulo rn.This converts the Hermitian dual-containing condition into a disjointness test on defining-set indices.
- Cyclotomic-coset criteria: Nontrivial codes exist exactly when at least one conjugate-reciprocal polynomial pair occurs among the irreducible factors of X^n−λ.The equivalent cyclotomic-coset criterion is that some C_e differs from C_−qe.
- Arithmetic criteria: For a+b≥2, existence follows if q≡1 (mod 4), if q≡−1 (mod 4) with a+b>e, or if the stated x_j conditions hold.The x_j conditions are that some x_j is zero, or all are nonzero with at least one x_j≥2.
- Examples and corollaries: The criteria show nonexistence for length 27 over F121 and length 5 over F34, but existence for length 10 over F34.The examples apply Theorems 2.11 and 2.12 to factorizations of rn and congruence/order conditions.
3 New quantum MDS codes
The paper derives new quantum MDS codes from dual-containing constacyclic MDS codes over Fq2. It treats several divisibility cases and also constructs codes of length q^2+1.
- The Hermitian construction reduces quantum MDS-code construction to finding dual-containing classical MDS constacyclic codes over Fq2.A dual-containing code satisfies C⊥H ⊆ C.
- Length q^2−1, divisor 7: For odd q with 7 | (q + 1), the paper derives another family of quantum MDS codes of length q^2−1 and gives examples for q = 13 and q = 27.The associated defining-set construction is dual-containing.
- Length q^2+1: When q = 10m + 3 or q = 10m + 7, the paper constructs quantum MDS codes of length q^2+1 with odd minimum distance d satisfying 3 ≤ d ≤ 4m + 1.The underlying constacyclic code has parameters [n, n − 4m, 4m + 1] and is dual-containing.
4 Summary and code comparisons
The paper lists known quantum MDS-code parameters, compares them with the newly constructed codes, and reports that the new codes can have larger minimum distance.
- The section first lists parameters of known quantum MDS codes before comparing them with the newly derived codes.
- Four classes of q-ary quantum MDS codes are constructed, and the paper compares their lengths with previously available classes.For fixed odd prime power p, only Class 3 is identified as having the possibility to reach the listed lengths.
- The examples show that the new codes have minimum distance greater than those available in the literature.