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Codes in general are often denoted by the letter C, and a code of length n and of rank k (i.e., having n code words in its basis and k rows in its generating matrix) is generally referred to as an (n, k) code. Linear block codes are frequently denoted as [n, k, d] codes, where d refers to the code's minimum Hamming distance between any two code ...
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. [jargon]
To show the existence of the linear code that satisfies those constraints, the probabilistic method is used to construct the random linear code. Specifically, the linear code is chosen by picking a generator matrix G ∈ F q k × n {\displaystyle G\in \mathbb {F} _{q}^{k\times n}} whose entries are randomly chosen elements of F q {\displaystyle ...
In mathematical terms, Hamming codes are a class of binary linear code. For each integer r ≥ 2 there is a code-word with block length n = 2 r − 1 and message length k = 2 r − r − 1 .
Linear block codes are summarized by their symbol alphabets (e.g., binary or ternary) and parameters (n,m,d min) [5] where n is the length of the codeword, in symbols, m is the number of source symbols that will be used for encoding at once, d min is the minimum hamming distance for the code. There are many types of linear block codes, such as
The Reed–Solomon code is a [n, k, n − k + 1] code; in other words, it is a linear block code of length n (over F) with dimension k and minimum Hamming distance = + The Reed–Solomon code is optimal in the sense that the minimum distance has the maximum value possible for a linear code of size ( n , k ); this is known as the Singleton bound .
Reed–Muller codes are linear block codes that are locally testable, locally decodable, and list decodable. These properties make them particularly useful in the design of probabilistically checkable proofs .
In coding theory, Hamming(7,4) is a linear error-correcting code that encodes four bits of data into seven bits by adding three parity bits. It is a member of a larger family of Hamming codes, but the term Hamming code often refers to this specific code that Richard W. Hamming introduced in 1950.