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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.).
Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥. This means that a codeword c is in C if and only if the matrix-vector product Hc ⊤ = 0 (some authors would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations.
Matrix digital rain. Matrix digital rain, or Matrix code, is the computer code featured in the Matrix series. The falling green code is a way of representing the activity of the simulated reality environment of the Matrix on screen by kinetic typography. All four Matrix movies, as well as the spin-off The Animatrix episodes, open with the code.
In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form . The dimension of C and its dual always add up to the length n : A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code.
Hadamard codes are obtained from an n-by-n Hadamard matrix H. In particular, the 2n codewords of the code are the rows of H and the rows of −H. To obtain a code over the alphabet {0,1}, the mapping −1 ↦ 1, 1 ↦ 0, or, equivalently, x ↦ (1 − x)/2, is applied to the matrix elements.
Linear code. In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. [1] Linear codes allow for more efficient encoding and ...
The RM(0, m) code is the repetition code of length N =2 m and weight N = 2 m−0 = 2 m−r. By 1 (,) = and has weight 1 = 2 0 = 2 m−r. The article bar product (coding theory) gives a proof that the weight of the bar product of two codes C 1, C 2 is given by
The BCH code with and higher has the generator polynomial. This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. It is also denoted as: (15, 1) BCH code. In fact, this code has only two codewords: 000000000000000 and 111111111111111 (a trivial repetition code ).