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Concept# Binary erasure channel

Summary

In coding theory and information theory, a binary erasure channel (BEC) is a communications channel model. A transmitter sends a bit (a zero or a one), and the receiver either receives the bit correctly, or with some probability P_e receives a message that the bit was not received ("erased") .
Definition
A binary erasure channel with erasure probability P_e is a channel with binary input, ternary output, and probability of erasure P_e. That is, let X be the transmitted random variable with alphabet {0,1}. Let Y be the received variable with alphabet {0,1,\text{e} }, where \text{e} is the erasure symbol. Then, the channel is characterized by the conditional probabilities:
:\begin{align}
\operatorname {Pr} [ Y = 0 | X = 0 ] &= 1 - P_e \
\operatorname {Pr} [ Y = 0 | X = 1 ] &= 0 \
\operatorname {Pr} [ Y = 1 | X = 0 ] &= 0 \
\operatorname {Pr} [ Y = 1 | X = 1

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COM-404: Information theory and coding

The mathematical principles of communication that govern the compression and transmission of data and the design of efficient methods of doing so.

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Coding theory

Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data

Communication channel

A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and compute

Information theory

Information theory is the mathematical study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in

In this paper we consider the finite-length performance of turbo codes in the waterfall region assuming the transmission on the binary erasure channel. We extend the finite-length scaling law of LDPC codes [1] and of Repeat-Accumulate codes [3] to this ensemble and we derive the scaling parameter. The obtained performance estimations match very well with numerical results.

In this paper we consider the finite-length performance of Repeat-Accumulate codes in the waterfall region assuming the transmission on the binary erasure channel. We extend the finite-length scaling law of LDPC codes [1] to this ensemble and we derive the scaling parameter. The obtained performance estimations match very well with numerical results.

Shannon, in his landmark 1948 paper, developed a framework for characterizing the fundamental limits of information transmission. Among other results, he showed that reliable communication over a channel is possible at any rate below its capacity. In 2008, Arikan discovered polar codes; the only class of explicitly constructed low-complexity codes that achieve the capacity of any binary-input memoryless symmetric-output channel. Arikan's polar transform turns independent copies of a noisy channel into a collection of synthetic almost-noiseless and almost-useless channels. Polar codes are realized by sending data bits over the almost-noiseless channels and recovering them by using a low-complexity successive-cancellation (SC) decoder, at the receiver. In the first part of this thesis, we study polar codes for communications. When the underlying channel is an erasure channel, we show that almost all correlation coefficients between the erasure events of the synthetic channels decay rapidly. Hence, the sum of the erasure probabilities of the information-carrying channels is a tight estimate of the block-error probability of polar codes when used for communication over the erasure channel. We study SC list (SCL) decoding, a method for boosting the performance of short polar codes. We prove that the method has a numerically stable formulation in log-likelihood ratios. In hardware, this formulation increases the decoding throughput by 53% and reduces the decoder's size about 33%. We present empirical results on the trade-off between the length of the CRC and the performance gains in a CRC-aided version of the list decoder. We also make numerical comparisons of the performance of long polar codes under SC decoding with that of short polar codes under SCL decoding. Shannon's framework also quantifies the secrecy of communications. Wyner, in 1975, proposed a model for communications in the presence of an eavesdropper. It was shown that, at rates below the secrecy capacity, there exist reliable communication schemes in which the amount of information leaked to the eavesdropper decays exponentially in the block-length of the code. In the second part of this thesis, we study the rate of this decay. We derive the exact exponential decay rate of the ensemble-average of the information leaked to the eavesdropper in Wyner's model when a randomly constructed code is used for secure communications. For codes sampled from the ensemble of i.i.d. random codes, we show that the previously known lower bound to the exponent is exact. Our ensemble-optimal exponent for random constant-composition codes improves the lower bound extant in the literature. Finally, we show that random linear codes have the same secrecy power as i.i.d. random codes. The key to securing messages against an eavesdropper is to exploit the randomness of her communication channel so that the statistics of her observation resembles that of a pure noise process for any sent message. We study the effect of feedback on this approximation and show that it does not reduce the minimum entropy rate required to approximate a given process. However, we give examples where variable-length schemes achieve much larger exponents in this approximation in the presence of feedback than the exponents in systems without feedback. Upper-bounding the best exponent that block codes attain, we conclude that variable-length coding is necessary for achieving the improved exponents.