On Optimal Nonlinear Systematic Codes
Most bounds on the size of codes hold for any code, whether linear or not. Notably, the Griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes. In this paper, we identify code parameters (q, d, k), namely, field size, minimum distance, and combinat...
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| Veröffentlicht in: | IEEE transactions on information theory 2016-06, Vol.62 (6), p.3103-3112 |
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| creator | Guerrini, Eleonora Meneghetti, Alessio Sala, Massimiliano |
| description | Most bounds on the size of codes hold for any code, whether linear or not. Notably, the Griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes. In this paper, we identify code parameters (q, d, k), namely, field size, minimum distance, and combinatorial dimension, for which the Griesmer bound also holds in the (systematic) nonlinear case. Moreover, we show that the Griesmer bound does not necessarily hold for a systematic code by explicit construction of a family of optimal systematic binary codes. On the other hand, we are able to provide some versions of the Griesmer bound holding for all the systematic codes. |
| doi_str_mv | 10.1109/TIT.2016.2553142 |
| format | Article |
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Notably, the Griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes. In this paper, we identify code parameters (q, d, k), namely, field size, minimum distance, and combinatorial dimension, for which the Griesmer bound also holds in the (systematic) nonlinear case. Moreover, we show that the Griesmer bound does not necessarily hold for a systematic code by explicit construction of a family of optimal systematic binary codes. 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Notably, the Griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes. In this paper, we identify code parameters (q, d, k), namely, field size, minimum distance, and combinatorial dimension, for which the Griesmer bound also holds in the (systematic) nonlinear case. Moreover, we show that the Griesmer bound does not necessarily hold for a systematic code by explicit construction of a family of optimal systematic binary codes. On the other hand, we are able to provide some versions of the Griesmer bound holding for all the systematic codes.</description><subject>Binary codes</subject><subject>Branch & bound algorithms</subject><subject>Coding theory</subject><subject>Combinatorial analysis</subject><subject>Combinatorics</subject><subject>Computer Science</subject><subject>Construction</subject><subject>Decoding</subject><subject>Error correction codes</subject><subject>Information Theory</subject><subject>Linear codes</subject><subject>Mathematics</subject><subject>Nonlinear systems</subject><subject>Nonlinearity</subject><subject>Optimization</subject><subject>Parameter identification</subject><subject>Systematics</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNpdkE1Lw0AQhhdRsFbvgpeACIKk7uxXkmMpagvFHqznZZNMcEs-6m5a6L93S0oPnoaB5x3eeQi5BzoBoNnrerGeMApqwqTkINgFGYGUSZwpKS7JiFJI40yI9JrceL8Jq5DARuRp1UarbW8bU0efXVvbFo2Lvg6-x8b0tohmXYn-llxVpvZ4d5pj8v3-tp7N4-XqYzGbLuOCS9nHphSClSwv0BTMlFlCqeAS8pSFKqoqM-RAS4F5rgoulALKKoUGpQKONGF8TF6Guz-m1lsXWrmD7ozV8-lS19Y1jabAmUyB7yHQzwO9dd3vDn2vG-sLrGvTYrfzGlImRRqe5gF9_Iduup1rwy8akkxIphhXgaIDVbjOe4fVuQNQfbSsg2V9tKxPlkPkYYhYRDzjyVGukPwPDKp0yQ</recordid><startdate>20160601</startdate><enddate>20160601</enddate><creator>Guerrini, Eleonora</creator><creator>Meneghetti, Alessio</creator><creator>Sala, Massimiliano</creator><general>IEEE</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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| subjects | Binary codes Branch & bound algorithms Coding theory Combinatorial analysis Combinatorics Computer Science Construction Decoding Error correction codes Information Theory Linear codes Mathematics Nonlinear systems Nonlinearity Optimization Parameter identification Systematics |
| title | On Optimal Nonlinear Systematic Codes |
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