On Base Field of Linear Network Coding
For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class N of multicast networks and obtain an explicit formula for the linear...
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| Vydáno v: | IEEE transactions on information theory Ročník 62; číslo 12; s. 7272 - 7282 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York
IEEE
01.12.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class N of multicast networks and obtain an explicit formula for the linear solvability of these networks, which involves the associated coset numbers of a multiplicative subgroup in a base field. The concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size. It further facilitates us to unveil infinitely many new multicast networks linearly solvable over GF(q) but not over GF(q') with q <; q', based on a subgroup order criterion. In particular: 1) for every k ≥ 2, an instance in N can be found linearly solvable over GF(2 2k ) but not over GF(2 2k+1 ) and 2) for arbitrary distinct primes p and p', there are infinitely many k and k' such that an instance in N can be found linearly solvable over GF(p k ) but not over GF(p' k ') with p k <; p' k' . |
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| AbstractList | For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class N of multicast networks and obtain an explicit formula for the linear solvability of these networks, which involves the associated coset numbers of a multiplicative subgroup in a base field. The concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size. It further facilitates us to unveil infinitely many new multicast networks linearly solvable over GF(q) but not over GF(q') with q <; q', based on a subgroup order criterion. In particular: 1) for every k ≥ 2, an instance in N can be found linearly solvable over GF(2 2k ) but not over GF(2 2k+1 ) and 2) for arbitrary distinct primes p and p', there are infinitely many k and k' such that an instance in N can be found linearly solvable over GF(p k ) but not over GF(p' k ') with p k <; p' k' . For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class N of multicast networks and obtain an explicit formula for the linear solvability of these networks, which involves the associated coset numbers of a multiplicative subgroup in a base field. The concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size. It further facilitates us to unveil infinitely many new multicast networks linearly solvable over GF( q ) but not over GF( q' ) with q<q', based on a subgroup order criterion. In particular: 1) for every k≥2 , an instance in N can be found linearly solvable over GF( 2^sup 2k^ ) but not over GF( 2^sup 2k+1^ ) and 2) for arbitrary distinct primes p and p' , there are infinitely many k and k'such that an instance in N can be found linearly solvable over GF( p^sup k& ) but not over GF( p'^sup k'^) with p^sup k^<p^sup k'^. |
| Author | Zongpeng Li Sun, Qifu Tyler Li, Shuo-Yen Robert |
| Author_xml | – sequence: 1 givenname: Qifu Tyler surname: Sun fullname: Sun, Qifu Tyler email: qfsun@ustb.edu.cn organization: Sch. of Comput. & Commun. Eng., Univ. of Sci. & Technol. Beijing, Beijing, China – sequence: 2 givenname: Shuo-Yen Robert surname: Li fullname: Li, Shuo-Yen Robert email: bobli@uestc.edu.cn organization: Univ. of Electron. Sci. & Technol. of China, Chengdu, China – sequence: 3 surname: Zongpeng Li fullname: Zongpeng Li email: zongpeng@ucalgary.ca organization: Dept. of Comput. Sci., Univ. of Calgary, Calgary, AB, Canada |
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| SubjectTerms | Algebra Communications networks Computers coset Cryptography Electronic mail Encoding generalized Cauchy-Davenport theorem Indexes multicast network multiplicative subgroup Network coding Receivers Sun |
| Title | On Base Field of Linear Network Coding |
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