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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 words. (The [n, k, d] notation should not be confused with the (n, M, d) notation used to denote a non-linear code of length n, size M (i.e., having M code words), and minimum Hamming distance d.)
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 computer networking, linear network coding is a program in which intermediate nodes transmit data from source nodes to sink nodes by means of linear combinations. Linear network coding may be used to improve a network's throughput, efficiency, and scalability , as well as reducing attacks and eavesdropping.
In the mathematics of coding theory, the Griesmer bound, named after James Hugo Griesmer, is a bound on the length of linear binary codes of dimension k and minimum distance d. There is also a very similar version for non-binary 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. [jargon]
Each transmitted code word in a constant-weight code is designed such that every code word that contains some positive or negative levels also contains enough of the opposite levels, such that the average level over each code word is zero. Examples of constant-weight codes include Manchester code and Interleaved 2 of 5. Use a paired disparity ...
[4] [5] In the case where C is a linear subspace of its Hamming space, it is called a linear code. [4] A typical example of linear code is the Hamming code. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes ...
Type II codes are binary self-dual codes which are doubly even. Type III codes are ternary self-dual codes. Every codeword in a Type III code has Hamming weight divisible by 3. Type IV codes are self-dual codes over F 4. These are again even. Codes of types I, II, III, or IV exist only if the length n is a multiple of 2, 8, 4, or 2 respectively.