"골레이 코드 (Golay code)"의 두 판 사이의 차이

수학노트
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1번째 줄: 1번째 줄:
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==개요==
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* [24,12,8] 골레이 코드 <math>C</math>
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* 유한체 <math>\mathbb{F}_2</math>위에 정의되는 선형코드 <math>C\subset \mathbb{F}_2^{24}</math>
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* 12차원 벡터 공간을 이루며, <math>C</math>의 원소의 개수는 <math>2^{12}=4096</math>
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* 가장 작은 길이를 갖는 코드는 길이 8
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* self-dual
  
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==codeword==
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===weight enumerator===
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* <math>W_{C}(x.y)=x^{24}+759 x^{16} y^8+2576 x^{12} y^{12}+759 x^8 y^{16}+y^{24}</math>
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* [[맥윌리엄스 항등식 (MacWilliams Identity)]]에 의해 다음이 성립
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:<math>
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W_{C}(x,y)=W_{C}\left(\frac{x+y}{\sqrt{2}},\frac{x-y}{\sqrt{2}}\right)
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</math>
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===길이 8인 코드===
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* 759개
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* [[슈타이너 시스템 S(5, 8, 24)]]으로 불린다
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[[파일:슈타이너 시스템 S(5, 8, 24)1.png]]
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==역사==
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* [[수학사 연표]]
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==관련된 항목들==
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* [[리치 격자(Leech lattice)]]
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* [[해밍코드(Hamming codes)]]
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* [[슈타이너 시스템 S(5, 8, 24)]]
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==매스매티카 파일 및 계산 리소스==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxY21Hc2Q3X25rbzQ/edit
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* http://mathworld.wolfram.com/GolayCode.html
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== 노트 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q1534522 Q1534522]
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===말뭉치===
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# Recently, some table-lookup decoding algorithms (TLDAs) have been used to correct the binary Golay code.<ref name="ref_340e953f">[https://www.sciencedirect.com/science/article/pii/S1665642313715438 High-Speed Decoding of the Binary Golay Code]</ref>
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# The problem of complete decoding the binary Golay code over error-and-erasure memoryless channels is addressed.<ref name="ref_248f9b19">[https://link.springer.com/article/10.1007/BF02997777 A decoding algorithm for the (23, 12, 7) golay code with error and erasure correction]</ref>
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# In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications.<ref name="ref_a8e218ba">[https://en.wikipedia.org/wiki/Binary_Golay_code Binary Golay code]</ref>
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# conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit).<ref name="ref_a8e218ba" />
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# Witt in 1938 published a construction of the largest Mathieu group that can be used to construct the extended binary Golay code.<ref name="ref_a8e218ba" />
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# 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.<ref name="ref_a8e218ba" />
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# It has been shown in the worksheet how to implement encoding and decoding of triple error correcting (24, 12) binary Golay code.<ref name="ref_d06c6eba">[https://www.maplesoft.com/applications/view.aspx?SID=1757&view=html Extended (24, 12) Binary Golay Code: Encoding and Decoding Procedures]</ref>
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# The automorphism group of the binary Golay code is the Mathieu group M 24 M_{24} , and the other Mathieu group are obtained as stabilisers of various sets in the Golay code.<ref name="ref_70ee6593">[https://ncatlab.org/nlab/show/binary+Golay+code binary Golay code in nLab]</ref>
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# If we delete any one component of the vectors in the extended Golay code, we obtain the perfect binary Golay code, a 12-dimensional subspace \(W’ \subset \mathbb{F}_2^{23}\).<ref name="ref_d5754dda">[https://blogs.ams.org/visualinsight/2015/12/01/golay-code/ Golay Code]</ref>
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# The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions.<ref name="ref_f2f5796f">[https://link.springer.com/article/10.1007/s10623-018-0532-z The sextuply shortened binary Golay code is optimal]</ref>
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# In the current computer-aided study, it is shown that \(A(18,8)=64\) and \(A(19,8)=128\), so an optimal code is obtained even after shortening the extended binary Golay code six times.<ref name="ref_f2f5796f" />
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# We provide a counterexample to show that the extended binary Golay code is not 1-perfect for the proposed poset block structures.<ref name="ref_b70cf7bd">[https://www.aimsciences.org/article/doi/10.3934/amc.2018037 Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces]</ref>
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# Two Golay codes exist—the 23-bit binary Golay code and the 11-trit ternary Golay code.<ref name="ref_8077d169">[https://royalsocietypublishing.org/doi/10.1098/rspa.2020.0187 Magic state distillation with the ternary Golay code]</ref>
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===소스===
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<references />
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q1534522 Q1534522]
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===Spacy 패턴 목록===
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* [{'LOWER': 'binary'}, {'LOWER': 'golay'}, {'LEMMA': 'code'}]

2021년 2월 17일 (수) 03:59 기준 최신판

개요

  • [24,12,8] 골레이 코드 \(C\)
  • 유한체 \(\mathbb{F}_2\)위에 정의되는 선형코드 \(C\subset \mathbb{F}_2^{24}\)
  • 12차원 벡터 공간을 이루며, \(C\)의 원소의 개수는 \(2^{12}=4096\)
  • 가장 작은 길이를 갖는 코드는 길이 8
  • self-dual


codeword

weight enumerator

\[ W_{C}(x,y)=W_{C}\left(\frac{x+y}{\sqrt{2}},\frac{x-y}{\sqrt{2}}\right) \]

길이 8인 코드

슈타이너 시스템 S(5, 8, 24)1.png



역사



관련된 항목들


매스매티카 파일 및 계산 리소스


노트

위키데이터

말뭉치

  1. Recently, some table-lookup decoding algorithms (TLDAs) have been used to correct the binary Golay code.[1]
  2. The problem of complete decoding the binary Golay code over error-and-erasure memoryless channels is addressed.[2]
  3. In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications.[3]
  4. conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit).[3]
  5. Witt in 1938 published a construction of the largest Mathieu group that can be used to construct the extended binary Golay code.[3]
  6. 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.[3]
  7. It has been shown in the worksheet how to implement encoding and decoding of triple error correcting (24, 12) binary Golay code.[4]
  8. The automorphism group of the binary Golay code is the Mathieu group M 24 M_{24} , and the other Mathieu group are obtained as stabilisers of various sets in the Golay code.[5]
  9. If we delete any one component of the vectors in the extended Golay code, we obtain the perfect binary Golay code, a 12-dimensional subspace \(W’ \subset \mathbb{F}_2^{23}\).[6]
  10. The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions.[7]
  11. In the current computer-aided study, it is shown that \(A(18,8)=64\) and \(A(19,8)=128\), so an optimal code is obtained even after shortening the extended binary Golay code six times.[7]
  12. We provide a counterexample to show that the extended binary Golay code is not 1-perfect for the proposed poset block structures.[8]
  13. Two Golay codes exist—the 23-bit binary Golay code and the 11-trit ternary Golay code.[9]

소스

  1. ↑ High-Speed Decoding of the Binary Golay Code
  2. ↑ A decoding algorithm for the (23, 12, 7) golay code with error and erasure correction
  3. ↑ 3.0 3.1 3.2 3.3 Binary Golay code
  4. ↑ Extended (24, 12) Binary Golay Code: Encoding and Decoding Procedures
  5. ↑ binary Golay code in nLab
  6. ↑ Golay Code
  7. ↑ 7.0 7.1 The sextuply shortened binary Golay code is optimal
  8. ↑ Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces
  9. ↑ Magic state distillation with the ternary Golay code

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'binary'}, {'LOWER': 'golay'}, {'LEMMA': 'code'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=골레이_코드_(Golay_code)&oldid=52191"