2007 Fiscal Year Final Research Report Summary
Study on the optimal linear codes problem
| Project/Area Number |
17540129
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Osaka Prefecture University |
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Principal Investigator |
MARUTA Tatsuya Osaka Prefecture University, Graduate School of Science, Professor (80239152)
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| Project Period (FY) |
2005 – 2007
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| Keywords | linear codes / error-correcting codes / extension of linear codes / Griesmer bound / optimal linear codes / projective geometry |
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| Research Abstract |
The optimal linear codes problem consisting of the following two research problems is one of the fundamental problems in coding theory.
(1) Find new linear codes, which can correct more errors than the known codes.
(2) Prove or disprove the existence of very good codes (e. g. the Griesmer codes attaining the Griesmer bound) .
The following research problem is also important to tackle the above problems.
(3) Find new extension theorems for linear codes. For instance, extension theorems can be applied to find new codes from old ones and to prove the non-existence of some kind of Griesmer codes.
In this study, we first investigate the extendibility of ternary linear codes in detail. Then, employing the geometrical methods we have developed there, we found new extension theorems for quaternary linear codes and for 3-weight (mod q) codes over the field of order q.
For the problems (1) and (2) , we tackled the problem to find the minimum length n for which a code of length n, dimension k, and the minimum Hamming distance d over the field of order q for given q and k for all d. We have solved the problem especially for the cases (q, k) = (3, 6) and (5, 5) . Furthermore, we have determined the minimum length n for general (q, k) for some range of d. As for the problem (1) , we also did the research to find new codes with the aid of computers and we have found a lot of new linear codes over small fields.
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[Presentation] On optimal linear codes over F_52007
Author(s)
M. Shinohara, T. Maruta, A. Kikui
Organizer
8th International Conference on Finite Fields and Applications (Fq8) in Melbourne
Place of Presentation
Australia
Year and Date
20070700
Description
「研究成果報告書概要(欧文)」より
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