Advancing Error-Correcting Codes Through Algebraic Innovations

The recent developments in the research area of coding theory have shown a significant shift towards exploring novel algebraic structures and their applications in constructing advanced error-correcting codes. A notable trend is the utilization of group algebras and their properties to develop quantum error-correcting codes, leveraging the CSS construction for specific cases like semisimple dihedral group algebras. Additionally, there is a growing interest in two-dimensional constacyclic codes, which offer improved minimum distances compared to traditional cyclic codes, and in additive codes over chain rings, which promise optimal performance under the homogeneous metric. The field is also witnessing advancements in the construction of DNA codes and non-binary quantum codes, with record-breaking parameters achieved through innovative approaches involving quasi-cyclic codes and group ring codes. Furthermore, the study of Galois LCD codes and LCPs of codes over mixed alphabets is providing new insights into code enumeration and security applications. Overall, the research is progressing towards more complex and efficient code constructions, with a strong emphasis on algebraic methods and their practical implications in quantum and DNA coding.

Sources

Codes in algebras of direct products of groups and associated quantum codes

Two-dimensional Constacyclic Codes over $\mathbb{F}_q$

On Eisenstein additive codes over chain rings and linear codes over mixed alphabets

On Galois LCD codes and LCPs of codes over mixed alphabets

DNA codes from $(\text{\textbaro}, \mathfrak{d}, \gamma)$-constacyclic codes over $\mathbb{Z}_4+\omega\mathbb{Z}_4$

Codes from $A_m$-invariant polynomials

Some New Non-binary Quantum Codes from One-generator Quasi-cyclic Codes

Quantum Codes from Group Ring Codes

Additive codes attaining the Griesmer bound

On the structure of the Schur squares of Twisted Generalized Reed-Solomon codes and application to cryptanalysis

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