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 codes, and Reed–Muller codes. 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 may also refer to any error-correcting code that acts on a block of bits of input data to produce bits of output data . 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 . Here, is a finite and nonempty set and and 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 . The size of the alphabet is often written as . If , then the block code is called a binary block code. In many applications it is useful to consider to be a prime power, and to identify with the finite field . The message length k Messages are elements of , that is, strings of