Reed Muller codes Reed-Muller codes form impotrant class of cyclic multiple-error-correction codes. For any integers p and r , with , there exists a binary RM code of the r -th order, denoted as RM( r , p ), with the following parameters: block-length: , length of information word: , minimum dista...
Shortened linear code If number of information bits required to encode M messages, the smallest integer l that satisfies inequality , is less than the given code information part length k , then an linear code can be shortened to code, where . This can be done by omitting j information bits. They c...
Single parity check code A single-parity-check code uses a single-parity bit to generate codeword with an even or odd parity, i.e. an even or odd number of nonzero bits respectively. Then, and for an even-parity code, parity-check sum: Parity-check matrix and generator matrix are trivial: Decode...
T ransmission of encoded information through DMC Source messages are encoded in M codewords : where the codeword letter is a letter of input alphabet : , In DMC, noise sequences (error patterns) : are added to codewords, producing output sequences : where the output sequence letter is a letter o...
Variable length binary codes Codeword length . Average number of code letters (bits) per source symbol ( average length of a codeword): , for binary code: . If then, By proper selection of codeword lengths, the minimum distance of one bit from absolute minimum (4.2.2) can be achieved. Distance ...
An introduction to the graph theory A graph is a set of points, together with a set of arcs that connect pairs of these points. It can be more than one arc connecting the same pair of points. An arc can connect a point to itself, forming a loop. A graph can be described by giving the set: V = { v 1...
Binary relation Definition Let A and B be nonempty sets. A binary relation from A to B is a subset of the Cartesian Product A × B . A binary relation from A to A is called a binary relation on A . Let R ⊂ A × B ( R is a binary relation from A to B ). If ( a, b ) ∈ R , we say: a is related to b and ...
Boolean function and conjunctive normal form. A Boolean function p ( x 1 , x 2 , ..., x n ) is said to be in conjunctive normal form , if it is the conjunction of finite number of distinct terms, each of which has the form e 1 ∨ e 2 ∨ ... ∨ e n , where e i = x i or e i = x ′ i for all i = 1 , 2 , ,...
Boolean function and disjunctive normal form. Let x 1 , x 2 , ..., x n be a sequence of statement variables. Then any compound statement of these variables will be called a Boolean polynomial or Boolean function . p ( x 1 , x 2 , ..., x n ) - Boolean polynomial of variables x 1 , x 2 , ..., x n . 0...
Tautology and contradiction Theorem Let A and B be compound statements. If the compound statement A and A = ⇒ B are tautologies, then so is the compound statement B . The substitution theorem for tautologies Let A be a tautology and suppose that A contains the distinct statement variables p 1 , p 2...
