scholarly journals On the Hyper-Kloosterman Codes Over Galois Rings

2020 ◽  
Vol 12 (2) ◽  
pp. 12
Author(s):  
Etienne TANEDJEU ASSONGMO
Keyword(s):  
Finite Fields ◽  
Linear Code ◽  
Hamming Weight ◽  
Galois Rings

The Hyper-Kloosterman code was first defined over finite fields by Chinen-Hiramatsu, see (Chinen, & Hiramatsu, 2001). In the present paper we define the Hyper-Kloosterman codes over Galois rings R(pe;m). We show that this code is the trace of linear code over R(pe;m). By the Hyper-Kloostermann sums over Galois rings, we determine the Hamming weight of any codeword of this code over Galois rings.

Hamming weight distributions of multi-twisted codes over finite fields

2021 ◽  
Author(s):  
Varsha Chauhan ◽  
Anuradha Sharma ◽  
Sandeep Sharma ◽  
Monika Yadav
Keyword(s):  
Finite Fields ◽  
Hamming Weight ◽  
Weight Distributions

scholarly journals Low-rank parity-check codes over Galois rings

2020 ◽  
Author(s):  
Julian Renner ◽  
Alessandro Neri ◽  
Sven Puchinger
Keyword(s):  
Finite Fields ◽  
Failure Probability ◽  
Low Rank ◽  
Galois Rings ◽  
Parity Check ◽  
Gabidulin Codes ◽  
Rank Metric Codes ◽  
Small Failure Probability ◽  
Free Rank ◽  
Parity Check Codes

AbstractLow-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (IEEE Trans Inf Theory 65(12):7718–7735, 2019), we define and study LRPC codes over Galois rings—a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.’s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.

Finite Fields and Galois Rings

10.1142/8250 ◽  
2011 ◽  
Author(s):  
Zhe-Xian Wan
Keyword(s):  
Finite Fields ◽  
Galois Rings

scholarly journals Three authentication schemes without secrecy over finite fields and Galois rings

2021 ◽  
Author(s):  
Juan Carlos Ku-Cauich ◽  
Miguel Angel Márquez-Hidalgo
Keyword(s):  
Finite Fields ◽  
Gray Map ◽  
Galois Rings ◽  
Main Achievement ◽  
The Third ◽  
Key Space ◽  
Authentication Schemes ◽  
The Given ◽  
Message Space

We give three new authentication schemes without secrecy. The first two on finite fields and Galois rings, using Gray map for this link. The third construction is given on Galois rings. The main achievement in this work is to obtain optimal impersonation and substitution probabilities in the schemes. Additionally, in the first and second scheme, we simplify the source space and bring a better relationship between the size of the message space and the key space than the given in [8]. Finally, we provide a third scheme on Galois rings, which generalizes the scheme over finite fields constructed in [9].

scholarly journals Matrix powers over finite fields

1992 ◽  
Vol 15 (4) ◽  
pp. 767-771
Author(s):  
Maria T. Acosta-De-Orozco ◽  
Javier Gomez-Calderon
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Related Problem ◽  
Galois Rings

LetGF(q)denote the finite field of orderq=pewithpodd. LetMdenote the ring of2×2matrices with entries inGF(q). Letndenote a divisor ofq−1and assume2≤nand4does not dividen. In this paper, we consider the problem of determining the number ofn-th roots inMof a matrixB∈M. Also, as a related problem, we consider the problem of lifting the solutions ofX2=Bover Galois rings.

scholarly journals On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings

2009 ◽  
Vol 3 (4) ◽  
pp. 409-420 ◽  
Author(s):  
Sergio R. López-Permouth ◽  
◽  
Steve Szabo
Keyword(s):  
Hamming Weight ◽  
Galois Rings ◽  
Negacyclic Codes
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