Binary Golay Code
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
electronics engineering Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current f ...
, a binary Golay code is a type of linear
error-correcting code In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is ...
used in
digital communication Data transmission and data reception or, more broadly, data communication or digital communications is the transfer and reception of data in the form of a digital bitstream or a digitized analog signal transmitted over a point-to-point or ...
s. The binary Golay code, along with the
ternary Golay code In coding theory, the ternary Golay codes are two closely related error-correcting codes. The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distan ...
, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics. These codes are named in honor of Marcel J. E. Golay whose 1949 paper introducing them has been called, by
E. R. Berlekamp Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley.Contributors, ''IEEE Transactions on Information Theory'' 42, #3 (May 1996), p. 1048. DO10 ...
, the "best single published page" in coding theory. There are two closely related binary Golay codes. The extended binary Golay code, ''G''24 (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected. The other, the perfect binary Golay code, ''G''23, 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 A parity bit, or check bit, is a bit added to a string of binary code. Parity bits are a simple form of error detecting code. Parity bits are generally applied to the smallest units of a communication protocol, typically 8-bit octets (bytes) ...
). In standard coding notation the codes have parameters 4, 12, 8and 3, 12, 7 corresponding to the length of the codewords, the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of the code, and the minimum
Hamming distance In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...
between two codewords, respectively.


Mathematical definition

In mathematical terms, the extended binary Golay code ''G''24 consists of a 12-dimensional linear subspace ''W'' of the space of 24-bit words such that any two distinct elements of ''W'' differ in at least 8 coordinates. ''W'' is called a linear code because it is a vector space. In all, ''W'' comprises elements. * The elements of ''W'' are called ''
code words ''Code Words'' is an online publication about computer programming produced by the Recurse Center retreat community. It began publishing in December 2014, and has a quarterly schedule. The journal features original work by participants at the Cent ...
''. They can also be described as subsets of a set of 24 elements, where addition is defined as taking the symmetric difference of the subsets. * In the extended binary Golay code, all code words have
Hamming weight The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string o ...
s of 0, 8, 12, 16, or 24. Code words of weight 8 are called octads and code words of weight 12 are called dodecads. * Octads of the code ''G''24 are elements of the S(5,8,24)
Steiner system 250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) ...
. There are octads and 759 complements thereof. It follows that there are dodecads. * Two octads intersect (have 1's in common) in 0, 2, or 4 coordinates in the binary vector representation (these are the possible intersection sizes in the subset representation). An octad and a dodecad intersect at 2, 4, or 6 coordinates. * Up to relabeling coordinates, ''W'' is unique. The binary Golay code, ''G''23 is a
perfect code In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of pack ...
. That is, the spheres of radius three around code words form a partition of the vector space. ''G''23 is a 12-dimensional subspace of the space F. The automorphism group of the perfect binary Golay code ''G''23 (meaning the subgroup of the group ''S23'' of permutations of the coordinates of F which leave ''G''23 invariant), is the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
M_. The
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the extended binary Golay code is the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
M_, of order . M_ is transitive on octads and on dodecads. The other Mathieu groups occur as stabilizers of one or several elements of ''W''.


Constructions

*
Lexicographic code Lexicographic codes or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by Vladimir Levenshtein and by John Horton Conway and Neil Sloane. The binary lexicographic codes are ...
: 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. Then ''W'' can be defined as the span of ''w''1, ..., ''w''12. *
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
: Witt in 1938 published a construction of the largest Mathieu group that can be used to construct the extended binary Golay code. *
Quadratic residue code A quadratic residue code is a type of cyclic code. Examples Examples of quadratic residue codes include the (7,4) Hamming code over GF(2), the (23,12) binary Golay code over GF(2) and the (11,6) ternary Golay code over GF(3). Constructions There ...
: Consider the set ''N'' of quadratic non-residues (mod 23). This is an 11-element subset of the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
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 In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detecti ...
: The perfect G23 code can be constructed via the factorization of x^-1 over the binary field
GF(2) (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with ...
: :::x^ - 1 = (x-1)(x^ + x^9+x^7+x^6+x^5+x+1)(x^+x^+x^6+x^5+x^4+x^2+1). :It is the code generated by \left(x^+x^+x^6+x^5+x^4+x^2+1\right). Either of degree 11 irreducible factors can be used to generate the code. * Turyn's construction of 1967, "A Simple Construction of the Binary Golay Code," that starts from the Hamming code of length 8 and does not use the quadratic residues mod 23. * From the Steiner System S(5,8,24), consisting of 759 subsets of a 24-set. If one interprets the support of each subset as a 0-1-codeword of length 24 (with Hamming-weight 8), these are the "octads" in the binary Golay code. The entire Golay code can be obtained by repeatedly taking the
symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...
s of subsets, i.e. binary addition. An easier way to write down the Steiner system resp. the octads is the
Miracle Octad Generator In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for manipulating the Mathieu groups, binary Golay code and Leech lattice. Description The Miracle Octad Generator is a 4x6 array of combinatio ...
of R. T. Curtis, that uses a particular 1:1-correspondence between the 35 partitions of an 8-set into two 4-sets and the 35 partitions of the finite vector space \mathbb_2^4 into 4 planes. Nowadays often the compact approach of Conway's hexacode, that uses a 4×6 array of square cells, is used. * Winning positions in the
mathematical game A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games ne ...
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. * A
generator matrix In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. The codewords are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix. Terminol ...
for the binary Golay code is I A, where I is the 12×12 identity matrix, and A is the complement of the
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...
of the
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
.


A convenient representation

It is convenient to use the "
Miracle Octad Generator In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for manipulating the Mathieu groups, binary Golay code and Leech lattice. Description The Miracle Octad Generator is a 4x6 array of combinatio ...
" format, with co-ordinates in an array of 4 rows, 6 columns. Addition is taking the symmetric difference. All 6 columns have the same parity, which equals that of the top row. A partition of the 6 columns into 3 pairs of adjacent ones constitutes a trio. This is a partition into 3 octad sets. A subgroup, the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associat ...
PSL(2,7) x S3 of a trio subgroup of M24 is useful for generating a basis. PSL(2,7) permutes the octads internally, in parallel. S3 permutes the 3 octads bodily. The basis begins with octad T: :0 1 1 1 1 1 :1 0 0 0 0 0 :1 0 0 0 0 0 :1 0 0 0 0 0 and 5 similar octads. The sum N of all 6 of these code words consists of all 1's. Adding N to a code word produces its complement. Griess (p. 59) uses the labeling: :∞ 0 , ∞ 0 , ∞ 0 :3 2 , 3 2 , 3 2 :5 1 , 5 1 , 5 1 :6 4 , 6 4 , 6 4 PSL(2,7) is naturally the linear fractional group generated by (0123456) and (0∞)(16)(23)(45). The 7-cycle acts on T to give a subspace including also the basis elements :0 1 1 0 1 0 :0 0 0 0 0 0 :0 1 0 1 0 1 :1 1 0 0 0 0 and :0 1 1 0 1 0 :0 1 0 1 0 1 :1 1 0 0 0 0 :0 0 0 0 0 0 The resulting 7-dimensional subspace has a 3-dimensional quotient space upon ignoring the latter 2 octads. There are 4 other code words of similar structure that complete the basis of 12 code words for this representation of W. W has a subspace of dimension 4, symmetric under PSL(2,7) x S3, spanned by N and 3 dodecads formed of subsets , , and .


Practical applications of Golay codes


NASA deep space missions

Error correction was vital to data transmission in the Voyager 1 and 2 spacecraft particularly because memory constraints dictated offloading data virtually instantly leaving no second chances. Hundreds of color pictures of
Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but ...
and
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in their 1979, 1980, and 1981 fly-bys would be transmitted within a constrained telecommunications bandwidth. Color image transmission required three times as much data as black and white images, so the 7-error correcting
Reed–Muller code Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication. Moreover, the proposed 5G standard relies on the closely related polar codes for error correction in ...
that had been used to transmit the black and white Mariner images was replaced with the much higher data rate Golay (24,12,8) code.


Radio communications

The
MIL-STD-188 MIL-STD-188 is a series of U.S. military standards relating to telecommunications. Purpose Faced with "past technical deficiencies in telecommunications systems and equipment and software…that were traced to basic inadequacies in the applicat ...
American military standards for automatic link establishment in
high frequency High frequency (HF) is the ITU designation for the range of radio frequency electromagnetic waves (radio waves) between 3 and 30 megahertz (MHz). It is also known as the decameter band or decameter wave as its wavelengths range from one to ten ...
radio systems specify the use of an extended (24,12) Golay code for
forward error correction In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is ...
.


See also

*
Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
*
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 ...


References


Sources

* * * * * * * * {{CCSDS Error detection and correction