Search results
Results from the WOW.Com Content Network
Consider the input code as 1101 1110 0001 0110 (this is the previous code with one error). We know the degree of the polynomial p x {\textstyle p_{x}} is at most r = 2 {\textstyle r=2} , we start by searching for monomial of degree 2.
The Hadamard code is a linear code, and all linear codes can be generated by a generator matrix. This is a matrix such that Had ( x ) = x ⋅ G {\displaystyle {\text{Had}}(x)=x\cdot G} holds for all x ∈ { 0 , 1 } k {\displaystyle x\in \{0,1\}^{k}} , where the message x {\displaystyle x} is viewed as a row vector and the vector-matrix product ...
A BCH code with = is called a narrow-sense BCH code.; A BCH code with = is called primitive.; The generator polynomial () of a BCH code has coefficients from (). In general, a cyclic code over () with () as the generator polynomial is called a BCH code over ().
In a polynomial code over () with code length and generator polynomial () of degree , there will be exactly code words. Indeed, by definition, p ( x ) {\displaystyle p(x)} is a code word if and only if it is of the form p ( x ) = g ( x ) ⋅ q ( x ) {\displaystyle p(x)=g(x)\cdot q(x)} , where q ( x ) {\displaystyle q(x)} (the quotient ) is of ...
The advantage of choosing a primitive polynomial as the generator for a CRC code is that the resulting code has maximal total block length in the sense that all 1-bit errors within that block length have different remainders (also called syndromes) and therefore, since the remainder is a linear function of the block, the code can detect all 2 ...
The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay, contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including low-density parity-check codes, turbo codes, and fountain codes.
Turbo coding is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit.
A generator matrix for a linear [,,]-code has format , where n is the length of a codeword, k is the number of information bits (the dimension of C as a vector subspace), d is the minimum distance of the code, and q is size of the finite field, that is, the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc.).