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Galois self-orthogonal MDS codes with large dimensions (2412.05011v1)

Published 6 Dec 2024 in cs.IT and math.IT

Abstract: Let $q=pm$ be a prime power, $e$ be an integer with $0\leq e\leq m-1$ and $s=\gcd(e,m)$. In this paper, for a vector $v$ and a $q$-ary linear code $C$, we give some necessary and sufficient conditions for the equivalent code $vC$ of $C$ and the extended code of $vC$ to be $e$-Galois self-orthogonal. From this, we directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be $e$-Galois self-orthogonal. Furthermore, for all possible $e$ satisfying $0\leq e\leq m-1$, we classify them into three cases (1) $\frac{m}{s}$ odd and $p$ even; (2) $\frac{m}{s}$ odd and $p$ odd; (3) $\frac{m}{s}$ even, and construct several new classes of $e$-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our $e$-Galois self-orthogonal MDS codes can have dimensions greater than $\lfloor \frac{n+pe-1}{pe+1}\rfloor$, which are not covered by previously known ones. Moreover, by propagation rules, we obtain some new MDS codes with Galois hulls of arbitrary dimensions. As an application, many quantum codes can be obtained from these MDS codes with Galois hulls.

Summary

  • The paper establishes necessary and sufficient conditions for e-Galois self-orthogonality in GRS and EGRS codes.
  • It classifies all values of e and constructs new families of MDS codes that surpass traditional dimension thresholds.
  • The propagation rules developed enable the derivation of enhanced quantum error-correcting codes from classical MDS codes.

Insights into Galois Self-Orthogonal MDS Codes with Large Dimensions

This paper focuses on the construction and analysis of Galois self-orthogonal Maximum Distance Separable (MDS) codes having large dimensions. The authors investigate conditions under which certain linear codes, particularly Generalized Reed-Solomon (GRS) and Extended GRS (EGRS) codes, exhibit e-Galois self-orthogonality, a property wherein a code is contained within its e-Galois dual space. This work expands the field of known MDS codes by introducing new classes that surpass previous dimensional constraints.

Key Contributions

The paper's principal contributions can be summarized as follows:

  1. Framework for Galois Self-Orthogonality: The authors provide necessary and sufficient conditions for a vector vv in conjunction with a q-ary linear code CC, such that the equivalent code and its extended form are e-Galois self-orthogonal. This is further extended to GRS and EGRS codes, offering a method to identify conditions for Galois self-orthogonality.
  2. Classification and Construction: All values of ee satisfying 0em10 \leq e \leq m-1 are categorized into three cases influenced by the parity of ss and pp. Based on these observations, several classes of e-Galois self-orthogonal MDS codes are constructed. Notably, these constructions enable code dimensions greater than the conventionally known thresholds.
  3. Propagation Rules and Quantum Applications: Utilizing propagation rules, the authors generate new MDS codes having Galois hulls of arbitrary dimensions. These developments have direct applications in the field of quantum error correction, allowing the derivation of numerous quantum codes from the newly constructed MDS codes.

Implications and Future Work

Practical Implications

The construction of MDS codes with Galois self-orthogonality has significant repercussions in quantum information theory, particularly in designing Entanglement-Assisted Quantum Error-Correcting Codes (EAQECCs). Since various quantum error-correcting codes are derived from classical MDS codes with certain orthogonality properties, the methods presented provide a broader toolkit for EAQECCs with enhanced parameters.

Theoretical Implications

Theoretically, by advancing the understanding of the Galois inner product's utilization, this paper generalizes prior results related to Euclidean and Hermitian orthogonality in codes. The expanded dimensional capabilities and flexible lengths of the new code families offer vital insights into vector spaces over finite fields and their orthogonal properties.

Conclusion

The paper expands the boundaries of known Galois self-orthogonal MDS codes by exploring the large dimensions accessible through strategic classifications and conditions. Future research can explore the applications of these expanded capabilities in both classical and quantum coding theory, examining further enhancements and optimizations in quantum error correction frameworks. The utility and versatility of these constructions herald the potential for both practical advancements and theoretical insights in error correction and related disciplines.

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