AK-ARBIS: An improved AK-MCS based on the adaptive radial-based importance sampling for small failure probability

2020 ◽  
Vol 82 ◽  
pp. 101891 ◽  
Author(s):  
Wanying Yun ◽  
Zhenzhou Lu ◽  
Xian Jiang ◽  
Leigang Zhang ◽  
Pengfei He
Keyword(s):  
Importance Sampling ◽  
Failure Probability ◽  
Small Failure Probability

scholarly journals Sequential decoding of a general classical-quantum channel

2013 ◽  
Vol 469 (2157) ◽  
pp. 20130259 ◽  
Author(s):  
Mark M. Wilde
Keyword(s):  
Quantum Measurement ◽  
Failure Probability ◽  
High Probability ◽  
Polar Codes ◽  
Sequential Decoding ◽  
Single Instance ◽  
Small Failure Probability ◽  
Classical Quantum ◽  
Union Bound ◽  
Sequential Measurements

Because a quantum measurement generally disturbs the state of a quantum system, one might think that it should not be possible for a sender and receiver to communicate reliably when the receiver performs a large number of sequential measurements to determine the message of the sender. We show here that this intuition is not true, by demonstrating that a sequential decoding strategy works well even in the most general ‘one-shot’ regime, where we are given a single instance of a channel and wish to determine the maximal number of bits that can be communicated up to a small failure probability. This result follows by generalizing a non-commutative union bound to apply for a sequence of general measurements. We also demonstrate two ways in which a receiver can recover a state close to the original state after it has been decoded by a sequence of measurements that each succeed with high probability. The second of these methods will be useful in realizing an efficient decoder for fully quantum polar codes, should a method ever be found to realize an efficient decoder for classical-quantum polar codes.

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.

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