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The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
Computing the position of a particular unique tuple or matrix in a de Bruijn sequence or torus is known as the de Bruijn decoding problem. Efficient O ( n log n ) {\displaystyle \color {Blue}O(n\log n)} decoding algorithms exist for special, recursively constructed sequences [ 17 ] and extend to the two-dimensional case.
Other problems for which the greedy algorithm gives a strong guarantee, but not an optimal solution, include Set cover; The Steiner tree problem; Load balancing [11] Independent set; Many of these problems have matching lower bounds; i.e., the greedy algorithm does not perform better than the guarantee in the worst case.
Decoding of binary Goppa codes is traditionally done by Patterson algorithm, which gives good error-correcting capability (it corrects all design errors), and is also fairly simple to implement. Patterson algorithm converts a syndrome to a vector of errors.
Lexicographic code: Order the vectors in V lexicographically (i.e., interpret them as unsigned 24-bit binary integers and take the usual ordering). Starting with w 0 = 0, define w 1, w 2, ..., w 12 by the rule that w n is the smallest integer which differs from all linear combinations of previous elements in at least eight coordinates.
In computer science and information theory, a Huffman code is a particular type of optimal prefix code that is commonly used for lossless data compression.The process of finding or using such a code is Huffman coding, an algorithm developed by David A. Huffman while he was a Sc.D. student at MIT, and published in the 1952 paper "A Method for the Construction of Minimum-Redundancy Codes".
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.)
If more than () / transmission errors occur, the receiver cannot uniquely decode the received word in general as there might be several possible codewords. One way for the receiver to cope with this situation is to use list decoding, in which the decoder outputs a list of all codewords in a certain radius.