"골레이 코드 (Golay code)"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
(사용자 2명의 중간 판 28개는 보이지 않습니다)
1번째 줄: 1번째 줄:
−
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 수학노트 원문주소</h5>
+
==개요==
 +
* [24,12,8] 골레이 코드 <math>C</math>
 +
* 유한체 <math>\mathbb{F}_2</math>위에 정의되는 선형코드 <math>C\subset \mathbb{F}_2^{24}</math>
 +
* 12차원 벡터 공간을 이루며, <math>C</math>의 원소의 개수는 <math>2^{12}=4096</math>
 +
* 가장 작은 길이를 갖는 코드는 길이 8
 +
* self-dual
  
−
 
 
  
−
 
+
==codeword==
 +
===weight enumerator===
 +
* <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>
 +
* [[맥윌리엄스 항등식 (MacWilliams Identity)]]에 의해 다음이 성립
 +
:<math>
 +
W_{C}(x,y)=W_{C}\left(\frac{x+y}{\sqrt{2}},\frac{x-y}{\sqrt{2}}\right)
 +
</math>
  
−
<h5>개요</h5>
+
===길이 8인 코드===
 +
* 759개
 +
* [[슈타이너 시스템 S(5, 8, 24)]]으로 불린다
  
−
 
+
[[파일:슈타이너 시스템 S(5, 8, 24)1.png]]
 +
  
−
 
+
  
−
<h5>Golay code</h5>
+
==역사==
−
 
 
−
# Clear[n, k, x, y, A, B, G, H, V, hw, md, W, Gex, Vex, hwex, mdex, Wex]<br> n := 23<br> k := 12<br> (*F2[k] is a set of all row vectors in k-tuples*)<br> F2[k_] := Tuples[{0, 1}, k]<br> (*matrix for systematic form representation*)<br> c5 := {1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0};<br> MG := Table[RotateRight[c5, n], {n, 0, 10}]<br> AppendTo[MG, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}];<br> A := Transpose[MG]
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5> </h5>
 
−
 
 
−
* http://mathworld.wolfram.com/GolayCode.html
 
−
 
 
−
# Clear[n, k, x, y, A, B, G, H, V, hw, md, W, Gex, Vex, hwex, mdex, Wex]<br> n := 23<br> k := 12<br> (*F2[k] is a set of all row vectors in k-tuples*)<br> F2[k_] := Tuples[{0, 1}, k]<br> (*matrix for systematic form representation*)<br> A := {{1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1}, {1, 0, 1, 0, 1, 1, 0, 1,<br>    1, 0, 0, 1}, {1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0}, {1, 0, 1, 1, 1,<br>    0, 1, 1, 0, 1, 0, 0}, {1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0}, {1, 1,<br>    0, 1, 0, 1, 1, 1, 0, 0, 0, 1}, {1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1,<br>    0}, {1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0}, {1, 1, 1, 0, 1, 0, 1, 0,<br>    0, 0, 1, 1}, {1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1}, {0, 1, 1, 1, 1,<br>    1, 1, 1, 1, 1, 1, 1}}
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<math>x^24 + 759 x^16 y^8 + 2576 x^12 y^12 + 759 x^8 y^16 + y^24</math>
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5>related links</h5>
 
−
 
 
−
 
 
−
 
 
−
<h5>역사</h5>
 
−
 
 
−
 
 
  
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
−
* [[수학사연표 (역사)|수학사연표]]
+
* [[수학사 연표]]
  
−
 
+
  
−
 
 
  
−
<h5>메모</h5>
+
==관련된 항목들==
 +
* [[리치 격자(Leech lattice)]]
 +
* [[해밍코드(Hamming codes)]]
 +
* [[슈타이너 시스템 S(5, 8, 24)]]
 +
  
−
 
+
==매스매티카 파일 및 계산 리소스==
  
−
* Math Overflow http://mathoverflow.net/search?q=
+
* https://docs.google.com/file/d/0B8XXo8Tve1cxY21Hc2Q3X25rbzQ/edit
−
 
+
* http://mathworld.wolfram.com/GolayCode.html
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5>관련된 항목들</h5>
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역</h5>
 
−
 
 
−
*  단어사전<br>
 
−
** http://translate.google.com/#en|ko|
 
−
** http://ko.wiktionary.org/wiki/
 
−
* 발음사전 http://www.forvo.com/search/
 
−
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
−
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
−
* [http://www.kss.or.kr/pds/sec/dic.aspx 한국통계학회 통계학 용어 온라인 대조표]
 
−
* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
−
* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5>사전 형태의 자료</h5>
 
−
 
 
−
* http://ko.wikipedia.org/wiki/
 
−
* http://en.wikipedia.org/wiki/
 
−
* [http://eom.springer.de/default.htm The Online Encyclopaedia of Mathematics]
 
−
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
−
* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations]
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5>리뷰논문, 에세이, 강의노트</h5>
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5>관련논문</h5>
 
−
 
 
−
* http://www.jstor.org/action/doBasicSearch?Query=
 
−
* http://www.ams.org/mathscinet
 
−
* http://dx.doi.org/
 
  
−
 
 
  
−
 
+
== 노트 ==
  
−
<h5>관련도서</h5>
+
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q1534522 Q1534522]
 +
===말뭉치===
 +
# 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>
 +
# 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>
 +
# 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>
 +
# conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit).<ref name="ref_a8e218ba" />
 +
# 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" />
 +
# 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" />
 +
# 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>
 +
# 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>
 +
# 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>
 +
# 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>
 +
# 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" />
 +
# 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>
 +
# 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>
 +
===소스===
 +
<references />
  
−
*  도서내검색<br>
+
==메타데이터==
−
** http://books.google.com/books?q=
+
===위키데이터===
−
** http://book.daum.net/search/contentSearch.do?query=
+
* ID : [https://www.wikidata.org/wiki/Q1534522 Q1534522]
 +
===Spacy 패턴 목록===
 +
* [{'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"