Application of Constacyclic codes to Quantum MDS Codes (1403.2499v1)

Abstract: 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.

• The paper introduces number-theoretic conditions for dual-containing constacyclic codes that facilitate the construction of superior quantum MDS codes.
• It constructs new q-ary quantum MDS codes by leveraging specific Hermitian dual-containing properties in constacyclic codes over F₍q²₎.
• The research advances quantum error correction by yielding codes with enhanced minimum distances and guiding future quantum communications studies.

Overview of Constacyclic Codes in Quantum MDS Codes

The paper "Application of Constacyclic Codes to Quantum MDS Codes" by Bocong Chen, San Ling, and Guanghui Zhang explores the utilization of constacyclic codes in the construction of quantum maximal-distance-separable (MDS) codes. Quantum MDS codes are a pivotal class of quantum codes known for achieving optimal error correction capabilities. The primary focus is on constructing q-ary quantum MDS codes, accomplished by identifying linear MDS codes over $F_{q^2}$ that satisfy specific Hermitian dual-containing conditions.

Key Contributions

The authors present a thorough investigation into identifying conditions under which nontrivial dual-containing constacyclic codes exist. Such codes are instrumental in constructing quantum MDS codes with enhanced minimum distances, exceeding those found in existing literature. The highlights of the paper include:

• Existence Conditions: The paper formulates number-theoretic conditions that must be fulfilled for the existence of dual-containing constacyclic codes, avoiding unnecessary attempts in code construction. Specifically, it is demonstrated that these codes exist when the order of the multiplicative elements $X \in F^*_{q^2}$ serves as a divisor of $q+1$.
• Construction of Quantum Codes: Several classes of dual-containing MDS constacyclic codes are constructed, yielding new quantum MDS codes with parameters superior to current known codes. Notably, the authors derive quantum MDS codes from four specific classes, characterized by parameters:
  • Lengths of $q^2-1$ with minimum distances constrained by $d > 2(q-2)$.
  • Length $\frac{q^2-1}{5}$, $\frac{q^2-1}{7}$, and $\frac{q+1}{10}$, with specific conditions on $q$.

The paper provides concrete numerical results and proofs of conditions under which these quantum MDS codes are obtainable. For example, new codes exhibit minimum distances greater than comparable existing codes, notably when $q > 11$, $q > 19$, or $q > 27$ depending on the class.

Implications and Future Work

This research contributes a pragmatic approach to constructing quantum error-correcting codes with dimensions surpassing established Hermitian thresholds. The results pave the way for more robust quantum communication systems, beneficial in areas such as quantum computing and cryptography where error correction is critical.

Furthermore, the paper suggests exploring additional algebraic structures and field extensions to derive even superior quantum codes. Future work might involve extending these methods to broader classes of constacyclic codes and exploring their applicability in multi-dimensional quantum systems. Additionally, investigating the practical implementations and performance of such quantum MDS codes in real-world quantum computational systems could substantiate their theoretical advancements.

In conclusion, the paper provides substantial progress in the field of quantum coding theory. Its contributions are expected to facilitate further advancements and innovations in error-correcting code constructions within quantum information science.