More MDS codes of non-Reed-Solomon type
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MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and establish some sufficient and necessary conditions for them being MDS. Notably, these codes differ from Reed-Solomon codes up to monomial equivalence. Additionally, we also explore the cases in which these codes are almost MDS or near MDS. Applying our main results, we determine the covering radii and deep holes of the dual codes associated with specific Roth-Lempel codes and discover an infinite family of (almost) optimally extendable codes with dimension three.
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Forward citations
Cited by 7 Pith papers
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A framework for constructing non-GRS MDS-NMDS codes from deep holes and its application
A framework constructs new non-GRS MDS-NMDS codes from deep holes, yielding three new families while recovering prior results with lower computational complexity.
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Generalized Roth-Lempel codes yield new Hermitian self-orthogonal NMDS families and quantum NMDS codes attaining the quantum Singleton bound minus one.
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The MDS or NMDS for Modified GRS codes with flexible hull dimensions and lengths
Constructs and characterizes MDS/NMDS modified GRS and extended GRS codes with flexible Euclidean and Hermitian hull dimensions and lengths.
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Non-RS type cyclic MDS codes over finite fields via cyclotomic field reduction
A cyclotomic field reduction method constructs non-RS cyclic MDS codes over finite fields by converting MDS verification to characteristic-zero minor checks, producing codes with flexible parameters.
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Non-RS type cyclic MDS codes over finite fields via cyclotomic field reduction
A construction of non-RS cyclic MDS codes over finite fields is obtained by norm reduction in cyclotomic fields, turning the MDS property into a characteristic-zero minor check and covering prior parameter sets.
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Two Families of Linear Codes Containing Non-GRS MDS Codes
Two new families of linear codes are built from modified GRS generator matrices, producing non-GRS MDS codes with derived parity-check matrices and self-duality conditions.
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MDS and NMDS Codes from the Extended Twisted Generalized Reed-Solomon Codes
Extended TGRS codes are built by adding three columns to TGRS generator matrices; conditions for MDS/AMDS/NMDS are derived and non-GRS nature is shown via Schur product.
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