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Decoding methods. In coding theory, decoding is the process of translating received messages into codewords of a given code. There have been many common methods of mapping messages to codewords. These are often used to recover messages sent over a noisy channel, such as a binary symmetric channel .
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 ...
Linear block codes have the property of linearity, i.e. the sum of any two codewords is also a code word, and they are applied to the source bits in blocks, hence the name linear block codes. There are block codes that are not linear, but it is difficult to prove that a code is a good one without this property.
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.
Here, codeword polynomial is an element of a linear code whose code words are polynomials that are divisible by a polynomial of shorter length called the generator polynomial. Every codeword polynomial can be expressed in the form c ( x ) = a ( x ) g ( x ) {\displaystyle c(x)=a(x)g(x)} , where g ( x ) {\displaystyle g(x)} is the generator ...
The codeword for that symbol is the string of "0"s and "1"s that records which half of the divides it fell on. This method was proposed in a later (in print) technical report by Fano (1949). Shannon–Fano codes are suboptimal in the sense that they do not always achieve the lowest possible expected codeword length, as Huffman coding does.
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 ).
Book cipher. The King James Bible, a highly available publication suitable for the book cipher. A book cipher is a cipher in which each word or letter in the plaintext of a message is replaced by some code that locates it in another text, the key . A simple version of such a cipher would use a specific book as the key, and would replace each ...