Journal of Advanced Navigation Technology (한국항행학회논문지)

The Korean Navigation Institute (한국항행학회)
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A Weight on Boolean Algebras for Cryptography and Error Correcting Codes

암호학 및 오류 수정 코드를 위한 부울 대수 가중치 연구

  • Yon, Yong-Ho (Innovation Center for Engineering Education, Mokwon University) ;
  • Kang, An-Na (Innovation Center for Engineering Education, Mokwon University)
  • 연용호 (목원대학교 공학교육혁신센터) ;
  • 강안나 (목원대학교 공학교육혁신센터)
  • Received : 2011.10.05
  • Accepted : 2011.10.30
  • Published : 2011.10.31
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Abstract

A sphere-packing problem is to find an arrangement of the spheres to fill as large area of the given space as possible, and covering problems are optimization problems which are dual problems to the packing problems. We generalize the concepts of the weight and the Hamming distance for a binary code to those of Boolean algebra. In this paper, we define a weight and a distance on a Boolean algebra and research some properties of the weight and the distance. Also, we prove the notions of the sphere-packing bound and the Gilbert-Varshamov bound on Boolean algebra.

Sphere-packing problem은 주어진 공간에 가능한 한 많은 구(sphere)를 채울 수 있는 배열을 찾는 문제이고 covering problem은 이에 쌍대적인 최적화의 문제로 코딩이론에 적용된다. 본 논문에서는 이진 코드이론에서의 가중치(weight)와 해밍거리(Hamming distance)에 대한 개념을 부울 대수(Boolean algebra)의 개념으로 일반화한다. 부울 대수에서의 가중치와 이를 이용하여 거리함수를 정의하고, 이들의 기본적인 성질들을 밝힌다. 또한, 부울 대수에서의 sphere-packing bound와 Gilbert-Varshamov bound의 정리를 증명한다.

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