Weight Enumerators and Cardinalities for Number-Theoretic Codes

The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number...

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Bibliographic Details
Published inIEEE transactions on information theory Vol. 68; no. 11; pp. 7165 - 7173
Main Author Nozaki, Takayuki
Format Journal Article
LanguageEnglish
Published New York IEEE 01.11.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects
Binary codes
Cardinality
Codes
Congruences
Deletion
Errors
Hamming weight
Indexes
Informatics
Insertion
insertion/deletion correcting code
Linear codes
non-binary Tenengolts’ code
number-theoretic code
Parity check codes
Rings (mathematics)
weight enumerator
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Summary:The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts' codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3184776