Decoding Algorithms of Twisted GRS Codes and Twisted Goppa Codes

In this paper, we use extended Euclid's algorithm to propose new decoding algorithms for two classes of maximum distance separable (MDS) twisted generalized Reed-Solomon (TGRS) codes of parameters <inline-formula> <tex-math notation="LaTeX">[n, n-t, t+1] </tex-math>&...

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Vydáno v:IEEE transactions on information theory Ročník 71; číslo 2; s. 1018 - 1027
Hlavní autoři: Sun, Huan, Yue, Qin, Jia, Xue, Li, Chengju
Médium: Journal Article
Jazyk:angličtina
Vydáno: IEEE 01.02.2025
Témata:
Codecs
Codes
Decoding
decoding algorithm
Linear codes
Mathematics
MDS code
Polynomials
Public key cryptography
Reed-Solomon codes
Sun
Technological innovation
Twisted generalized Reed-Solomon code
twisted Goppa code
ISSN:0018-9448, 1557-9654
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Popis
Shrnutí:In this paper, we use extended Euclid's algorithm to propose new decoding algorithms for two classes of maximum distance separable (MDS) twisted generalized Reed-Solomon (TGRS) codes of parameters <inline-formula> <tex-math notation="LaTeX">[n, n-t, t+1] </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">\Bbb F_{q} </tex-math></inline-formula>. For even t, the algorithms can correct <inline-formula> <tex-math notation="LaTeX">\frac {t}{2} </tex-math></inline-formula> errors with time complexity <inline-formula> <tex-math notation="LaTeX">O(qn) </tex-math></inline-formula>. Moreover, we also give a new decoding algorithm for a class of twisted Goppa codes. For even degree t of a Goppa polynomial, it can also correct <inline-formula> <tex-math notation="LaTeX">\frac {t}{2} </tex-math></inline-formula> errors, which generalizes a <inline-formula> <tex-math notation="LaTeX">\lfloor \frac {t-1}{2}\rfloor </tex-math></inline-formula>-error-correcting decoding algorithm by Sui and Yue (2023).
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2024.3509895