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<title>Binary Golay code</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Binary Golay code</span></span>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Extended binary Golay code</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"></span><div class="infobox-caption"><a href="Generator_matrix" title="Generator matrix">Generator matrix</a></div></td></tr><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><a href="Marcel_J._E._Golay" title="Marcel J. E. Golay">Marcel J. E. Golay</a></td></tr><tr><th colspan="2" class="infobox-header" style="background:#ccf;">Classification</th></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="Linear_block_code" class="mw-redirect" title="Linear block code">Linear block code</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_block_length_n" title="Block code">Block length</a></th><td class="infobox-data">24</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_message_length_k" title="Block code">Message length</a></th><td class="infobox-data">12</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_rate_R" title="Block code">Rate</a></th><td class="infobox-data">12/24 = 0.5</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_distance_d" title="Block code">Distance</a></th><td class="infobox-data">8</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_alphabet_Σ" title="Block code">Alphabet size</a></th><td class="infobox-data">2</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#Popular_notation" title="Block code">Notation</a></th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [24,12,8]_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle [24,12,8]_{2}}</annotation>

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</math></span><img src="./94468240ab43ea76dd2a763b3ae1442fdc6bdded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.228ex; height:2.843ex;" alt="{\displaystyle [24,12,8]_{2}}" loading="lazy"></span>-code</td></tr><tr><td colspan="2" class="infobox-navbar"><style data-mw-deduplicate="TemplateStyles:r1129693374">

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<table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Perfect binary Golay code</th></tr><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><a href="Marcel_J._E._Golay" title="Marcel J. E. Golay">Marcel J. E. Golay</a></td></tr><tr><th colspan="2" class="infobox-header" style="background:#ccf;">Classification</th></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="Linear_block_code" class="mw-redirect" title="Linear block code">Linear block code</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_block_length_n" title="Block code">Block length</a></th><td class="infobox-data">23</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_message_length_k" title="Block code">Message length</a></th><td class="infobox-data">12</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_rate_R" title="Block code">Rate</a></th><td class="infobox-data">12/23 ~ 0.522</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_distance_d" title="Block code">Distance</a></th><td class="infobox-data">7</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#The_alphabet_Σ" title="Block code">Alphabet size</a></th><td class="infobox-data">2</td></tr><tr><th scope="row" class="infobox-label"><a href="Block_code#Popular_notation" title="Block code">Notation</a></th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [23,12,7]_{2}}">

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Electronics_engineering" class="mw-redirect" title="Electronics engineering">electronics engineering</a>, a <b>binary Golay code</b> is a type of linear <a href="Error-correcting_code" class="mw-redirect" title="Error-correcting code">error-correcting code</a> used in <a href="Digital_communication" class="mw-redirect" title="Digital communication">digital communications</a>. The binary Golay code, along with the <a href="Ternary_Golay_code" title="Ternary Golay code">ternary Golay code</a>, has a particularly deep and interesting connection to the theory of <a href="Finite_sporadic_group" class="mw-redirect" title="Finite sporadic group">finite sporadic groups</a> in mathematics.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> These codes are named in honor of <a href="Marcel_J._E._Golay" title="Marcel J. E. Golay">Marcel J. E. Golay</a> whose 1949 paper<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> introducing them has been called, by <a href="E._R._Berlekamp" class="mw-redirect" title="E. R. Berlekamp">E. R. Berlekamp</a>, the "best single published page" in <a href="Coding_theory" title="Coding theory">coding theory</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>There are two closely related binary Golay codes. The <b>extended binary Golay code</b>, <i>G</i><sub>24</sub> (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 4-bit errors can be detected.

The other, the <b>perfect binary Golay code</b>, <i>G</i><sub>23</sub>, 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 <a href="Parity_bit" title="Parity bit">parity bit</a>). In standard coding notation, the codes have parameters [24, 12, 8] and [23, 12, 7], corresponding to the length of the codewords, the <a href="Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> of the code, and the minimum <a href="Hamming_distance" title="Hamming distance">Hamming distance</a> between two codewords, respectively.

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<div class="mw-heading mw-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2></div>

<p>In mathematical terms, the extended binary Golay code <i>G</i><sub>24</sub> consists of a 12-dimensional <a href="Linear_subspace" title="Linear subspace">linear subspace</a> <i>W</i> of the space <span class="nowrap"><i>V</i> = <b>F</b><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">24</sup><br><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span></span> of 24-bit words such that any two distinct elements of <i>W</i> differ in at least 8 coordinates. <i>W</i> is called a linear code because it is a vector space. In all, <i>W</i> comprises <span class="nowrap">4096 = 2<sup>12</sup></span> elements.

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<ul><li>The elements of <i>W</i> are called <i><a href="Code_word_(communication)" title="Code word (communication)">code words</a></i>. 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.</li>

<li>In the extended binary Golay code, all code words have <a href="Hamming_weight" title="Hamming weight">Hamming weights</a> of 0, 8, 12, 16, or 24. Code words of weight 8 are called <b>octads</b> and code words of weight 12 are called <b>dodecads</b>.</li>

<li>Octads of the code <i>G</i><sub>24</sub> are elements of the S(5,8,24) <a href="Steiner_system" title="Steiner system">Steiner system</a>. There are <span class="nowrap">759 = 3 × 11 × 23</span> octads and 759 complements thereof. It follows that there are <span class="nowrap">2576 = 2<sup>4</sup> × 7 × 23</span> dodecads.</li>

<li>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.</li>

<li>Up to relabeling coordinates, <i>W</i> is unique.</li></ul>

<p>The binary Golay code, <i>G</i><sub>23</sub> is a <a href="Perfect_code" class="mw-redirect" title="Perfect code">perfect code</a>. That is, the spheres of radius three around code words form a partition of the vector space. <i>G</i><sub>23</sub> is a 12-dimensional <a href="Linear_subspace" title="Linear subspace">subspace</a> of the space <b>F</b><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">23</sup><br><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span>.

</p><p>The automorphism group of the perfect binary Golay code <i>G</i><sub>23</sub> (meaning the subgroup of the group <i>S<sub>23</sub></i> of permutations of the coordinates of <b>F</b><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">23</sup><br><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span> which leave <i>G</i><sub>23</sub> invariant), is the <a href="Mathieu_group" title="Mathieu group">Mathieu group</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{23}}">

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</math></span><img src="./d764890eb74937322f8857127a5e7b89ba3cc6af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.13ex; height:2.509ex;" alt="{\displaystyle M_{23}}" loading="lazy"></span>. The <a href="Automorphism_group" title="Automorphism group">automorphism group</a> of the extended binary Golay code is the <a href="Mathieu_group" title="Mathieu group">Mathieu group</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{24}}">

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</p><p>There is a single word of weight 24, which is a 1-dimensional invariant subspace. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{24}}">

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</math></span><img src="./1dcb770e233d2a596badb4c413303daa17613099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.13ex; height:2.509ex;" alt="{\displaystyle M_{24}}" loading="lazy"></span> therefore has an 11-dimensional irreducible representation on the field with 2 elements. In addition, since the binary golay code is a 12-dimensional subspace of a 24-dimensional space, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{24}}">

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</math></span><img src="./1dcb770e233d2a596badb4c413303daa17613099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.13ex; height:2.509ex;" alt="{\displaystyle M_{24}}" loading="lazy"></span> also acts on the 12-dimensional <a href="Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient space</a>, called the <i>binary Golay cocode</i>. A word in the cocode is in the same <a href="Coset" title="Coset">coset</a> as a word of length 0, 1, 2, 3, or 4. In the last case, 6 (disjoint) cocode words all lie in the same coset. There is an 11-dimensional invariant subspace, consisting of cocode words with odd weight, which gives <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{24}}">

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<div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2></div>

<ul><li><a href="Lexicode" class="mw-redirect" title="Lexicode">Lexicographic code</a>: Order the vectors in <i>V</i> lexicographically (i.e., interpret them as unsigned 24-bit binary integers and take the usual ordering). Starting with <i>w</i><sub>0</sub> = 0, define <i>w</i><sub>1</sub>, <i>w</i><sub>2</sub>, ..., <i>w</i><sub>12</sub> by the rule that <i>w</i><sub><i>n</i></sub> is the smallest integer which differs from all linear combinations of previous elements in at least eight coordinates. Then <i>W</i> can be defined as the span of <i>w</i><sub>1</sub>, ..., <i>w</i><sub>12</sub>.</li>

<li><a href="Mathieu_group" title="Mathieu group">Mathieu group</a>: Witt in 1938 published a construction of the largest Mathieu group that can be used to construct the extended binary Golay code.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Quadratic_residue_code" title="Quadratic residue code">Quadratic residue code</a>: Consider the set <i>N</i> of quadratic non-residues (mod 23). This is an 11-element subset of the <a href="Cyclic_group" title="Cyclic group">cyclic group</a> <b>Z</b>/23<b>Z</b>. Consider the translates <i>t</i>+<i>N</i> of this subset. Augment each translate to a 12-element set <i>S</i><sub><i>t</i></sub> by adding an element ∞. Then labeling the basis elements of <i>V</i> by 0, 1, 2, ..., 22, ∞, <i>W</i> can be defined as the span of the words <i>S</i><sub><i>t</i></sub> together with the word consisting of all basis vectors. (The perfect code is obtained by leaving out ∞.)</li>

<li>As a <a href="Cyclic_code" title="Cyclic code">cyclic code</a>: The perfect G<sub>23</sub> code can be constructed via the factorization of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{23}+1}">

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</math></span><img src="./209ecf52f9685ac80bce51feb90cc0bec6ff268d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.209ex; height:2.843ex;" alt="{\displaystyle x^{23}+1}" loading="lazy"></span> over the binary field <a href="GF(2)" title="GF(2)">GF(2)</a>: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{23}+1=(x+1)(x^{11}+x^{9}+x^{7}+x^{6}+x^{5}+x+1)(x^{11}+x^{10}+x^{6}+x^{5}+x^{4}+x^{2}+1).}">

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<annotation encoding="application/x-tex">{\displaystyle x^{23}+1=(x+1)(x^{11}+x^{9}+x^{7}+x^{6}+x^{5}+x+1)(x^{11}+x^{10}+x^{6}+x^{5}+x^{4}+x^{2}+1).}</annotation>

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</math></span></span> It is the code generated by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x^{11}+x^{10}+x^{6}+x^{5}+x^{4}+x^{2}+1\right)}">

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</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x^{11}+x^{10}+x^{6}+x^{5}+x^{4}+x^{2}+1\right)}</annotation>

</semantics>

</math></span><img src="./17cc9b2552df918b10bd354fbf3ca1d3212fde89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.282ex; height:3.343ex;" alt="{\displaystyle \left(x^{11}+x^{10}+x^{6}+x^{5}+x^{4}+x^{2}+1\right)}" loading="lazy"></span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Either of degree 11 irreducible factors can be used to generate the code.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li>

<li>Turyn's construction of 1967, "A Simple Construction of the Binary Golay Code," that starts from the <a href="Hamming_code" title="Hamming code">Hamming code</a> of length 8 and does not use the quadratic residues mod 23.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>From the <a href="Steiner_system#The_Steiner_system_S.285.2C_8.2C_24.29" title="Steiner system">Steiner System S(5,8,24)</a>, 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 <a href="Symmetric_difference" title="Symmetric difference">symmetric differences</a> of subsets, i.e. binary addition. An easier way to write down the Steiner system resp. the octads is the <a href="Miracle_Octad_Generator" title="Miracle Octad Generator">Miracle Octad Generator</a> 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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}^{4}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}^{4}}</annotation>

</semantics>

</math></span><img src="./494231d102673e6f89527375aeec0be640b4330b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.475ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{2}^{4}}" loading="lazy"></span> into 4 planes.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Nowadays often the compact approach of Conway's hexacode, that uses a 4×6 array of square cells, is used.</li>

<li>Winning positions in the <a href="Mathematical_game" title="Mathematical game">mathematical game</a> 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.</li>

<li>A <a href="Generator_matrix" title="Generator matrix">generator matrix</a> for the binary Golay code is <b>I A</b>, where <b>I</b> is the 12×12 identity matrix, and <b>A</b> is the complement of the <a href="Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of the <a href="Icosahedron" title="Icosahedron">icosahedron</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="A_convenient_representation">A convenient representation</h3></div>

<p>It is convenient to use the "<a href="Miracle_Octad_Generator" title="Miracle Octad Generator">Miracle Octad Generator</a>" format, with coordinates 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.

</p><p>A partition of the 6 columns into 3 pairs of adjacent ones constitutes a <a href="Mathieu_group_M24#Trio_subgroup" title="Mathieu group M24">trio</a>. This is a partition into 3 octad sets. A subgroup, the <a href="Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear group</a> PSL(2,7) x S<sub>3</sub> of a trio subgroup of M<sub>24</sub> is useful for generating a basis. PSL(2,7) permutes the octads internally, in parallel. S<sub>3</sub> permutes the 3 octads bodily.

</p><p>The basis begins with octad T:

</p>

<pre>0 1 1 1 1 1

1 0 0 0 0 0

</pre>

<p>and 5 similar octads. The sum <b>N</b> of all 6 of these code words consists of all 1's. Adding N to a code word produces its complement.

</p><p>Griess (p. 59) uses the labeling:

</p>

<pre>∞ 0 | ∞ 0 | ∞ 0

3 2 | 3 2 | 3 2

5 1 | 5 1 | 5 1

6 4 | 6 4 | 6 4

</pre>

<p>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

</p>

<pre>0 1 1 0 1 0

0 0 0 0 0 0

0 1 0 1 0 1

1 1 0 0 0 0

</pre>

<p>and

</p>

<pre>0 1 1 0 1 0

0 1 0 1 0 1

1 1 0 0 0 0

0 0 0 0 0 0

</pre>

<p>The resulting 7-dimensional subspace has a 3-dimensional quotient space upon ignoring the latter 2 octads.

</p><p>There are 4 other code words of similar structure that complete the basis of 12 code words for this representation of W.

</p><p>W has a subspace of dimension 4, symmetric under PSL(2,7) x S<sub>3</sub>, spanned by N and 3 dodecads formed of subsets {0,3,5,6}, {0,1,4,6}, and {0,1,2,5}.

</p>

<div class="mw-heading mw-heading2"><h2 id="Practical_applications_of_Golay_codes">Practical applications of Golay codes</h2></div>

<div class="mw-heading mw-heading3"><h3 id="NASA_deep_space_missions">NASA deep space missions</h3></div>

<p>Error correction was vital to data transmission in the <a href="Voyager_program" title="Voyager program">Voyager</a> 1 and 2 spacecraft particularly because memory constraints dictated offloading data virtually instantly leaving no second chances. Hundreds of color pictures of <a href="Jupiter" title="Jupiter">Jupiter</a> and <a href="Saturn" title="Saturn">Saturn</a> 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 <a href="Reed%E2%80%93Muller_code" title="Reed–Muller code">Reed–Muller code</a> 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.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Radio_communications">Radio communications</h3></div>

<p>The <a href="MIL-STD-188" title="MIL-STD-188">MIL-STD-188</a> American military standards for <a href="Automatic_link_establishment" title="Automatic link establishment">automatic link establishment</a> in <a href="High_frequency" title="High frequency">high frequency</a> radio systems specify the use of an extended (24,12) Golay code for <a href="Forward_error_correction" class="mw-redirect" title="Forward error correction">forward error correction</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p><p>In <a href="Two-way_radio" title="Two-way radio">two-way radio</a> communication <a href="Squelch#DCS" title="Squelch">digital-coded squelch (DCS, CDCSS)</a> system uses 23-bit Golay (23,12) code word which has the ability to detect and correct errors of 3 or fewer bits.

</p>

<ul><li><a href="Leech_lattice" title="Leech lattice">Leech lattice</a></li>

<li><a href="Linear_code" title="Linear code">Linear code</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="reflist">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFThompson1983">Thompson 1983</a></span>

</li>

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</style><cite id="CITEREFGolay1949" class="citation journal cs1">Golay, Marcel J. E. (1949). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230410124137/https://www.lama.univ-savoie.fr/pagesmembres/hyvernat/Enseignement/2223/info602/TP-Golay/golay_paper.pdf">"Notes on Digital Coding"</a> <span class="cs1-format">(PDF)</span>. <i><a href="Proc._IRE" class="mw-redirect" title="Proc. IRE">Proc. IRE</a></i>. <b>37</b>: 657. Archived from <a rel="nofollow" class="external text" href="https://pierre-hyvernat.apps.math.cnrs.fr/data/Enseignement/2223/info602/TP-Golay/golay_paper.pdf">the original</a> <span class="cs1-format">(PDF)</span> on April 10, 2023.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFBerlekamp1974" class="citation cs2">Berlekamp, E. R. (1974), <i>Key Papers in the Development of Coding Theory</i>, I.E.E.E. Press, p. 4</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFHansen2011" class="citation journal cs1">Hansen, Robert Peter (2011). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://scholarworks.sjsu.edu/etd_theses/4053">"Construction and Simplicity of the Large Mathieu Groups"</a></span>. <i>Master's Theses</i>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.31979%2Fetd.qnhv-a5us">10.31979/etd.qnhv-a5us</a>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFRoman1996">Roman 1996</a>, p. 324 Example 7.4.3</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFPless1998">Pless 1998</a>, p. 114</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFTuryn1967">Turyn 1967</a>, Section VI</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFCullinane" class="citation web cs1">Cullinane, Steven H. <a rel="nofollow" class="external text" href="http://finitegeometry.org/sc/24/MOG.html">"The Miracle Octad Generator"</a>. <i>Finite Geometry of the Square and Cube</i>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFCherowitzo" class="citation web cs1">Cherowitzo, Bill. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130927082620/http://www-math.ucdenver.edu/~wcherowi/courses/m7409/mariner9talk.pdf">"Combinatorics in Space - The Mariner 9 Telemetry System"</a> <span class="cs1-format">(PDF)</span>. <a href="University_of_Colorado_Denver" title="University of Colorado Denver">University of Colorado Denver</a>. Archived from <a rel="nofollow" class="external text" href="http://www-math.ucdenver.edu/~wcherowi/courses/m7409/mariner9talk.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-09-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-06-06</span></span>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFJohnson1991" class="citation web cs1">Johnson, Eric E. (1991-02-24). <a rel="nofollow" class="external text" href="http://tracebase.nmsu.edu/hf/reports/Golay_Codec.pdf">"An Efficient Golay Codec for MIL-STD-188-141A and FED-STD-1045"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2017-12-09</span></span>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://everyspec.com/MIL-STD/MIL-STD-0100-0299/download.php?spec=MIL-STD-187_721B.026564.pdf">"Military Standard: Planning and Guidance Standard for Automated Control Applique for HF Radio"</a> <span class="cs1-format">(PDF)</span>. <i>EverySpec: Specifications, Standards, Handbooks and Mil-Spec documents</i>. 1994-04-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-12-09</span></span>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div>

<ul><li><cite id="CITEREFConwaySloane1999" class="citation cs2"><a href="John_Horton_Conway" title="John Horton Conway">Conway, John Horton</a>; <a href="Neil_Sloane" title="Neil Sloane">Sloane, Neil J. A.</a> (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=upYwZ6cQumoC"><i>Sphere Packings, Lattices and Groups</i></a>, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98585-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0920369">0920369</a></cite></li>

<li><cite id="CITEREFCurtis,_R._T.1976" class="citation journal cs1">Curtis, R. T. (1976). "A new combinatorial approach to M<sub>24</sub>". <i><a href="Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society" title="Mathematical Proceedings of the Cambridge Philosophical Society">Mathematical Proceedings of the Cambridge Philosophical Society</a></i>. <b>79</b> (1): <span class="nowrap">25–</span>42. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976MPCPS..79...25C">1976MPCPS..79...25C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100052075">10.1017/S0305004100052075</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122860631">122860631</a>.</cite></li>

<li><cite id="CITEREFGreferath,_Marcus2003" class="citation book cs1">Greferath, Marcus (2003). "Golay Codes". In Proakis, John G. (ed.). <i>Encyclopedia of Telecommunications</i>. Wiley. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F0471219282.eot371">10.1002/0471219282.eot371</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0471219282</bdi>.</cite></li>

<li><cite id="CITEREFGriess,_Robert_L.1998" class="citation book cs1"><a href="Robert_Griess" title="Robert Griess">Griess, Robert L.</a> (1998). <i>Twelve Sporadic Groups</i>. Springer. p. 167. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-62778-4</bdi>.</cite></li>

<li><cite id="CITEREFPless1998" class="citation cs2"><a href="Vera_Pless" title="Vera Pless">Pless, Vera</a> (1998), <a href="Introduction_to_the_Theory_of_Error-Correcting_Codes" title="Introduction to the Theory of Error-Correcting Codes"><i>Introduction to the Theory of Error-Correcting Codes</i></a> (3rd ed.), John Wiley & Sons, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-19047-9</bdi></cite></li>

<li><cite id="CITEREFRoman1996" class="citation cs2">Roman, Steven (1996), <i>Coding and Information Theory</i>, Graduate Texts in Mathematics #134, Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-97812-7</bdi></cite></li>

<li><cite id="CITEREFThompson1983" class="citation book cs1">Thompson, Thomas M. (1983). <i>From Error Correcting Codes through Sphere Packings to Simple Groups</i>. Carus Mathematical Monographs. Vol. 21. Mathematical Association of America. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-023-7</bdi>.</cite></li>

<li><cite id="CITEREFTuryn1967" class="citation report cs1">Turyn, Richard J.; et al. (1967). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181030114235/http://www.dtic.mil/dtic/tr/fulltext/u2/656783.pdf">Research to Develop the Algebraic Theory of Codes (Section VI)</a> <span class="cs1-format">(PDF)</span> (Report). Air Force Cambridge Research Laboratories. Archived from <a rel="nofollow" class="external text" href="http://www.dtic.mil/dtic/tr/fulltext/u2/656783.pdf">the original</a> <span class="cs1-format">(PDF)</span> on October 30, 2018.</cite></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Consultative_Committee_for_Space_Data_Systems222" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Consultative_Committee_for_Space_Data_Systems222" style="font-size:114%;margin:0 4em"><a href="Consultative_Committee_for_Space_Data_Systems" title="Consultative Committee for Space Data Systems">Consultative Committee for Space Data Systems</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Data compression</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Images

<li>Data

<ul><li><a href="Adaptive_coding" title="Adaptive coding">Adaptive Entropy Coder</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Error Correction</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<dl><dt>Current</dt>

<dd><a href="Concatenated_error_correction_code" title="Concatenated error correction code">Concatenated codes</a></dd>

<dd><a href="Turbo_code" title="Turbo code">Turbo codes</a></dd>

<dt>Proposed</dt>

<dd><a href="Low-density_parity-check_code" title="Low-density parity-check code">LDPC codes</a></dd></dl>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Telemetry command uplink</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Command-loss_timer" title="Command-loss timer">Command-loss timer</a></li>

<li><a href="Proximity-1_Space_Link_Protocol" title="Proximity-1 Space Link Protocol">Proximity-1 Space Link Protocol</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Telemetry downlink</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spacecraft_Monitoring_%26_Control" class="mw-redirect" title="Spacecraft Monitoring & Control">Spacecraft Monitoring & Control</a></li>

<li><a href="Beacon_mode_service" title="Beacon mode service">Beacon mode service</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Telemetry general</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Space_Communications_Protocol_Specifications" title="Space Communications Protocol Specifications">Space Communications Protocol Specifications</a> (SCPS): <a href="Performance_Enhancing_Proxy" class="mw-redirect" title="Performance Enhancing Proxy">Performance Enhancing Proxy</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Telemetry modulation systems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<dl><dt>Current</dt>

<dd><a href="OQPSK" class="mw-redirect" title="OQPSK">OQPSK</a></dd>

<dt>Proposed</dt>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Frequencies</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Networking, interoperability and monitoring</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Service-oriented_architecture" title="Service-oriented architecture">Service-oriented architecture</a> (<a href="Message_Abstraction_Layer" title="Message Abstraction Layer">Message Abstraction Layer</a>)</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-24" href="https://en.wikipedia.org/wiki/?title=Binary_Golay_code&oldid=1297075784">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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