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Polynomial code. In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length) that are divisible by a given fixed polynomial (of shorter length, called the generator polynomial ).
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 [1] would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. [2]
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 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).
The core of the full Aztec code. 40 bits are available between the orientation marks for encoding parameters. Message data is placed in a spiral pattern around the core. The mode message begins "01011100", indicating 01 2 +1 = 2 layers, and 011100 2 +1 = 29 data codewords (of 6 bits each). The compact Aztec code core may be surrounded by 1 to 4 ...
Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When r and m are integers with 0 ≤ r ≤ m, the Reed–Muller code with parameters r and m is denoted as RM (r, m). When asked to encode a message consisting of k bits, where holds, the RM (r, m) code produces a codeword consisting of ...
A negacyclic code is a constacyclic code with λ=-1. [8] A quasi-cyclic code has the property that for some s, any cyclic shift of a codeword by s places is again a codeword. [9] A double circulant code is a quasi-cyclic code of even length with s=2. [9] Quasi-twisted codes and multi-twisted codes are further generalizations of constacyclic ...