# Binary erasure channel

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### Communication channel

**channelchannelscommunications channel**

A binary erasure channel (or BEC) is a common communications channel model used in coding theory and information theory.

### Peter Elias

**EliasPeter and Marjorie (née Forbes) Elias**

The BEC was introduced by Peter Elias of MIT in 1955 as a toy example.

He also established the binary erasure channel and proposed list decoding of error-correcting codes as an alternative to unique decoding.

### Information theory

**information-theoreticinformation theoristinformation**

A binary erasure channel (or BEC) is a common communications channel model used in coding theory and information theory.

* A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel.

### Packet erasure channel

Closely related to the binary erasure channel is the packet erasure channel which shares many similar theoretical results with the binary erasure channel.

This channel model is closely related to the binary erasure channel.

### Binary symmetric channel

**BSCq-ary symmetric channel**

Unlike the binary symmetric channel, when the receiver gets a bit, it is 100% certain that the bit is correct.

Various explicit codes for achieving the capacity of the binary erasure channel have also come up recently.

### Deletion channel

The binary erasure channel should not be confused with a deletion channel where bits from the transmitter are either transmitted to the receiver (with probability p) or dropped without notifying the receiver (with probability 1-p).

The deletion channel should not be confused with the binary erasure channel which is much simpler to analyze.

### Landauer's principle

**Landauer limitLandauer's erasure principleprinciple**

disambiguation: Landauer's principle

### Coding theory

**algebraic coding theorycodingchannel code**

A binary erasure channel (or BEC) is a common communications channel model used in coding theory and information theory.

### Bit

**bitsbinary digitbinary digits**

In this model, a transmitter sends a bit (a zero or a one), and the receiver either receives the bit or it receives a message that the bit was not received ("erased").

### Binary number

**binarybinary numeral systembase 2**

The BEC is a binary channel; that is, it can transmit only one of two symbols (usually called 0 and 1).

### Noise (electronics)

**noiseelectronic noiseelectrical noise**

This channel is often used by theorists because it is one of the simplest noisy channels to analyze.

### Communication theory

**communicationcommunications theorycommunication theorist**

Many problems in communication theory can be reduced to a BEC.

### Reduction (complexity)

**reductionreducedreductions**

Many problems in communication theory can be reduced to a BEC.

### Random variable

**random variablesrandom variationrandom**

That is, let X be the transmitted random variable with alphabet \{0,1\}.

### Conditional probability

**conditional probabilitiesconditionalconditioned**

Then, the channel is characterized by the conditional probabilities

### Channel capacity

**capacitydata capacityinformation capacity**

The capacity of a BEC is 1-P_e.

### Erasure code

**erasure codingerasure correcting codeerasure**

* Erasure code

### Low-density parity-check code

**LDPCLDPC codelow-density parity-check codes**

LDPC codes are, which means that practical constructions exist that allow the noise threshold to be set very close (or even arbitrarily close on the binary erasure channel) to the theoretical maximum (the Shannon limit) for a symmetric memoryless channel.

### Sudoku Codes

Sudoku codes are non-linear forward error correcting codes following rules of sudoku puzzles designed for an erasure channel.

### Error detection and correction

**error correctionerror detectionerror-correction**

The latter approach is particularly attractive on an erasure channel when using a rateless erasure code.

### Luby transform code

**LT codesLT code**

They are erasure correcting codes because they can be used to transmit digital data reliably on an erasure channel.

### Reed–Solomon error correction

**Reed–Solomon codeReed-SolomonReed-Solomon error correction**

Specialized forms of Reed–Solomon codes, specifically Cauchy-RS and Vandermonde-RS, can be used to overcome the unreliable nature of data transmission over erasure channels.

### Polar code (coding theory)

**polar codespolar codepolar coding**

It typically involves modelling an encoder, a decoder, a channel (such as AWGN, BSC, BEC), and a code-construction module.

### Michael Luby

**LubyLuby, MichaelMichael G. Luby**

These were the first LDPC codes based on an irregular degree design that has proved crucial to all later good LDPC code designs, which provably achieve channel capacity for the erasure channel, and which have linear time encoding and decoding algorithms.