Binary Golay code

In mathematics and computer science, a binary Golay code is a type of 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. The code is named in honour of Marcel J. E. Golay.

There are two closely related binary Golay codes. The extended binary Golay code encodes 12 bits of data in a 24-bit word in such a way that any triple-bit error can be corrected and any quadruple-bit error can be detected. The other, the perfect binary Golay code, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit). In standard code notation the codes have parameters [24, 12, 8] and [23, 12, 7] .

Mathematical definition

In mathematical terms, the extended binary Golay code consists of a 12-dimensional subspace "W" of the space "V"=F224 of 24-bit words such that any two distinct elements of "W" differ in at least eight coordinates. Equivalently, any non-zero element of "W" has at least eight non-zero coordinates.
* The possible sets of non-zero coordinates as "w" ranges over "W" are called "code words". In the extended binary Golay code, all code words have Hamming weight 0, 8, 12, 16, or 24.
* Up to relabeling coordinates, "W" is unique.

The perfect binary Golay code is a perfect code; that is, the spheres of radius 3 around code words form a partition of the vector space.

The automorphism group of the binary Golay code is the Mathieu group M_{23}. The automorphism group of the extended binary Golay code is the Mathieu group M_{24}. The other Mathieu groups occur as stabilizers of one or several elements of "W".

The Golay code words of weight 8 are elements of the S(5,8,24) Steiner system.

Constructions

#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"1 = 0, define "w"2, "w"3, ..., "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. Then "W" can be defined as the span of "w"1, ..., "w"12.
#Quadratic residue code: Consider the set "N" of quadratic non-residues (mod 23). This is an 11-element subset of the cyclic group Z/23Z. Consider the translates "t"+"N" of this subset. Augment each translate to a 12-element set "S""t" by adding an element ∞. Then labeling the basis elements of "V" by 0, 1, 2, ..., 22, ∞, "W" can be defined as the span of the words "S""t" together with the word consisting of all basis vectors. (The perfect code is obtained by leaving out ∞.)
#As a Cyclic code: The perfect G23 code can be constructed via factorization of x^{23}-1, it is the code generated by x^{11}+x^{10}+x^6+x^5+x^4+x^2+1 / x^{23}-1
#The "Miracle Octad Generator" of R. T. Curtis: This uses a 4×6 array of square cells to picture the 759 Hamming-weight-8 code words, or "octads," of the extended binary Golay code. The remaining code words are obtained via symmetric differences of subsets of the 24 cells-- i.e., by binary addition.
#Winning positions in the mathematical game of Mogul: a position in Mogul is a row of 24 coins. Each turn consists of flipping from one to seven coins such that the leftmost of the flipped coins goes from head to tail. The losing positions are those with no legal move. If heads are interpreted as 1 and tails as 0 then moving to a codeword from the extended binary Golay code guarantees it will be possible to force a win.

Practical applications of Golay codes

NASA Deep Space Missions

The Voyager 1 & 2 spacecraft needed to transmit hundreds of color pictures of Jupiter and Saturn in their 1979 and 1980 fly-bys within a constrained telecommunications bandwidth.
* Color image transmission required 3 times the amount of data, so the Golay (24,12,8) code was used. [http://www-math.cudenver.edu/~wcherowi/courses/m6409/mariner9talk.pdf]
* This Golay code is only 3-error correcting, but it could be transmitted at a much higher data rate than the Hadamard code that was used during the Mariner mission.

ALE HF data communications

The new US government standards for automatic link establishment (ALE) in High Frequency (HF) radio systems specifies the use of an extended (24,12) Golay block code for forward error correction (FEC).

* The Extended (24,12) Golay Code specified is a (24,12) block code.
* This code encodes 12 data bits to produce 24-bit code words.
* It is furthermore a systematic code, meaning that the 12 data bits are present in unchanged form in the code word.

The minimum Hamming distance between any two code words (the number of bits by which any pair of code words differs) is 8, giving this code the power to detect up to 7 errors in each code word (and correct none), correct up to 3 while detecting 4, or any intermediate combination.
* These modes may be listed as (0,7), (1,6), (2,5), and (3,4), where the first number indicates the number of errors which may be corrected, and the second the number of errors which will be detected by each mode.

See also

* Mathieu group
* Steiner system
* Leech lattice

References

* Curtis, R. T. A new combinatorial approach to M24. Math. Proc. Camb. Phil. Soc. 79 (1976) 25-42.
* Griess, Robert L.: "Twelve Sporadic Groups", Springer-Verlag, 1998.
* Thompson, Thomas M.: "From Error Correcting Codes through Sphere Packings to Simple Groups", Carus Mathematical Monographs, Mathematical Association of America, 1983.


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