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Wikipedia / Block code

In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of all block codes in a unified way. Such limitations often take the form of bounds that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors.

Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay codess, Reed–Muller codes and Polar codess. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes, because they can be generated using Boolean polynomials.

Algebraic block codes are typically hard-decoded using algebraic decoders.

The term block code. Consequently, the block coder is a memoryless device. Under this definition codes such as turbo codes, terminated convolutional codes and other iteratively decodable codes (turbo-like codes) would also be considered block codes. A non-terminated convolutional encoder would be an example of a non-block (unframed) code, which has memory and is instead classified as a tree code.

This article deals with "algebraic block codes".

The block code and its parameters

Error-correcting codes are used to reliably transmit digital data over unreliable communication channels subject to channel noise. When a sender wants to transmit a possibly very long data stream using a block code, the sender breaks the stream up into pieces of some fixed size. Each such piece is called message and the procedure given by the block code encodes each message individually into a codeword, also called a block in the context of block codes. The sender then transmits all blocks to the receiver, who can in turn use some decoding mechanism to (hopefully) recover the original messages from the possibly corrupted received blocks. The performance and success of the overall transmission depends on the parameters of the channel and the block code.

Formally, a block code is an injective mapping

  1. .

is a finite and nonempty set are integers. The meaning and significance of these three parameters and other parameters related to the code are described below.

The alphabet Σ

The data stream to be encoded is modeled as a string over some alphabet, then the block code is called a binary to be a prime power with the finite field.

The message length k

is called the message length or dimension of a block code.

The block length n

The block length.

The rate R

The rate of a block code is defined as the ratio between its message length and its block length:

  1. .

measures the overhead that occurs due to the encoding with the block code. It is a simple information theoretical is an injective map.

The distance d

The distance or minimum distance of a block code is the minimum number of positions in which any two distinct codewords differ, and the relative distance denote the Hamming distance is defined as

  1. .

Since any code has to be injective. Besides, the distance equals the minimum weight for linear block codes because:

  1. .

A larger distance allows for more error correction and detection. For example, if we only consider errors that may change symbols of the sent codeword but never erase or add them, then the number of errors is the number of positions in which the sent codeword and the received word differ. A code with distance transmission errors occur, the receiver cannot uniquely decode the received word in general as there might be several possible codewords. One way for the receiver to cope with this situation is to use list decoding, in which the decoder outputs a list of all codewords in a certain radius.

Popular notation

, the index is sometimes dropped. For maximum distance separable codes-component may be missing.

.

Examples

As mentioned above, there are a vast number of error-correcting codes that are actually block codes. The first error-correcting code was the Hamming(7,4) code, developed by Richard W. Hamming code.

Reed–Solomon codes being a prime power. Rank codes. Hadamard codes.

Error detection and correction properties

, there is no other codeword in the Hamming ball-dimension words whose Hamming distance has the following properties:

Lower and upper bounds of block codes

Family of codes

is called family of codes.

Rate of family of codes

Relative distance of family of codes

, a set of lower and upper bounds of block codes are known.

Hamming bound

Singleton bound

The Singleton bound is that the sum of the rate and the relative distance of a block code cannot be much larger than 1:

  1. .

. Reed–Solomon codes are non-trivial examples of codes that satisfy the singleton bound with equality.

Plotkin bound

.

with distance :

For any

Gilbert–Varshamov bound

is the -ary entropy function.

Johnson bound

. be the maximum number of codewords in a Hamming ball of radius of distance .

Then we have the Johnson Bound

Elias–Bassalygo bound

Sphere packings and lattices

Block codes are tied to the sphere packing problem which has received some attention over the years. In two dimensions, it is easy to visualize. Take a bunch of pennies flat on the table and push them together. The result is a hexagon pattern like a bee's nest. But block codes rely on more dimensions which cannot easily be visualized. The powerful Golay code used in deep space communications uses 24 dimensions. If used as a binary code (which it usually is), the dimensions refer to the length of the codeword as defined above.

The theory of coding uses the N-dimensional sphere model. For example, how many pennies can be packed into a circle on a tabletop or in 3 dimensions, how many marbles can be packed into a globe. Other considerations enter the choice of a code. For example, hexagon packing into the constraint of a rectangular box will leave empty space at the corners. As the dimensions get larger, the percentage of empty space grows smaller. But at certain dimensions, the packing uses all the space and these codes are the so-called perfect codes. There are very few of these codes.

Another property is the number of neighbors a single codeword may have. Again, consider pennies as an example. First we pack the pennies in a rectangular grid. Each penny will have 4 near neighbors (and 4 at the corners which are farther away). In a hexagon, each penny will have 6 near neighbors. Respectively, in three and four dimensions, the maximum packing is given by the 12-face and 24-cell with 12 and 24 neighbors, respectively. When we increase the dimensions, the number of near neighbors increases very rapidly. In general, the value is given by the kissing numbers.

The result is that the number of ways for noise to make the receiver choose a neighbor (hence an error) grows as well. This is a fundamental limitation of block codes, and indeed all codes. It may be harder to cause an error to a single neighbor, but the number of neighbors can be large enough so the total error probability actually suffers.

See also

References