골레이 코드 (Golay code)

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

• \(W_{C}(x.y)=x^{24}+759 x^{16} y^8+2576 x^{12} y^{12}+759 x^8 y^{16}+y^{24}\)
• 맥윌리엄스 항등식 (MacWilliams Identity)에 의해 다음이 성립

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

길이 8인 코드

• 759개
• 슈타이너 시스템 S(5, 8, 24)으로 불린다

역사

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

관련된 항목들

• 리치 격자(Leech lattice)
• 해밍코드(Hamming codes)
• 슈타이너 시스템 S(5, 8, 24)

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxY21Hc2Q3X25rbzQ/edit
• http://mathworld.wolfram.com/GolayCode.html

노트

위키데이터

• ID : Q1534522

말뭉치

1. Recently, some table-lookup decoding algorithms (TLDAs) have been used to correct the binary Golay code.
2. The problem of complete decoding the binary Golay code over error-and-erasure memoryless channels is addressed.
3. In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications.
4. conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit).
5. Witt in 1938 published a construction of the largest Mathieu group that can be used to construct the extended binary Golay code.
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.
7. It has been shown in the worksheet how to implement encoding and decoding of triple error correcting (24, 12) binary Golay code.
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.
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 \(W’ \subset \mathbb{F}_2^{23}\).
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.
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.
12. We provide a counterexample to show that the extended binary Golay code is not 1-perfect for the proposed poset block structures.
13. Two Golay codes exist—the 23-bit binary Golay code and the 11-trit ternary Golay code.

메타데이터

위키데이터

• ID : Q1534522

Spacy 패턴 목록

• [{'LOWER': 'binary'}, {'LOWER': 'golay'}, {'LEMMA': 'code'}]