Search results
Results From The WOW.Com Content Network
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics.
Used in a football pool with 11 games, the ternary Golay code corresponds to 729 bets and guarantees exactly one bet with at most 2 wrong outcomes. The set of codewords with Hamming weight 5 is a 3- (11,5,4) design . The generator matrix given by Golay (1949, Table 1.) is.
The binary Golay code, independently developed in 1949, is an application in coding theory. More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting up to seven. It was used to communicate with the Voyager probes, as it is much more compact than the previously-used Hadamard code.
In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for studying the Mathieu groups, binary Golay code and Leech lattice. Description [ edit ] The Miracle Octad Generator is a 4x6 array of combinations describing any point in 24-dimensional space.
The 4096 codewords of the 24-bit binary Golay code are generated, and the 759 codewords with a Hamming weight of 8 correspond to the S(5,8,24) system. The Golay code can be constructed by many methods, such as generating all 24-bit binary strings in lexicographic order and discarding those that differ from some earlier one in fewer than 8 ...
Other examples of classical block codes include Golay, BCH, Multidimensional parity, and Hamming codes. Hamming ECC is commonly used to correct NAND flash memory errors. [6] This provides single-bit error correction and 2-bit error detection.
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.
A Barker code or Barker sequence is a finite sequence of N values of +1 and −1, with the ideal autocorrelation property, such that the off-peak (non-cyclic) autocorrelation coefficients. are as small as possible: for all . [1] Only nine Barker sequences [6] are known, all of length N at most 13. [7] Barker 's 1953 paper asked for sequences ...
But at certain dimensions, the packing uses all the space and these codes are the so-called "perfect" codes. The only nontrivial and useful perfect codes are the distance-3 Hamming codes with parameters satisfying (2 r – 1, 2 r – 1 – r, 3), and the [23,12,7] binary and [11,6,5] ternary Golay codes.
Examples of quadratic residue codes include the (,) Hamming code over (), the (,) binary Golay code over () and the (,) ternary Golay code over (). Constructions