Linear Size Constant-Composition Codes Meeting the Johnson Bound

The Johnson-type upper bound on the maximum size of a code of length n , distance d=2w-1 , and constant composition {\overline {w}} is \lfloor \dfrac {\vphantom {R^{.}}n}{w_{1}}\rfloor , where w is the total weight and w_{1} is the largest component of {\overline {w}} . Recently, Chee et a...

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Published in:IEEE transactions on information theory 2018-02, Vol.64 (2), p.909-917
Main Authors: Chee, Yeow Meng, Zhang, Xiande
Format: Article
Language:English
Subjects:
balanced packings
Binary codes
Binary system
Codes
Combinatorial analysis
Composition
Constant-composition codes
difference families
Frequency modulation
Indexes
Johnson-type bound
Lower bounds
Memoryless systems
Nickel
Upper bound
Upper bounds
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description The Johnson-type upper bound on the maximum size of a code of length n , distance d=2w-1 , and constant composition {\overline {w}} is \lfloor \dfrac {\vphantom {R^{.}}n}{w_{1}}\rfloor , where w is the total weight and w_{1} is the largest component of {\overline {w}} . Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let N_{ccc}({\overline {w}}) be the smallest such length. The determination of N_{ccc}({\overline {w}}) is trivial for binary codes. This paper provides a lower bound on N_{ccc}({\overline {w}}) , which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by the refining method, we determine the values of N_{ccc}({\overline {w}}) , for all q -ary constant-composition codes, provided that 3w_{1}\geq w with finite possible exceptions.
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Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula> be the smallest such length. The determination of <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula> is trivial for binary codes. This paper provides a lower bound on <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula>, which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by the refining method, we determine the values of <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula>, for all <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-ary constant-composition codes, provided that <inline-formula> <tex-math notation="LaTeX">3w_{1}\geq w </tex-math></inline-formula> with finite possible exceptions.]]></description><identifier>ISSN: 0018-9448</identifier><identifier>EISSN: 1557-9654</identifier><identifier>DOI: 10.1109/TIT.2017.2689026</identifier><identifier>CODEN: IETTAW</identifier><language>eng</language><publisher>New York: IEEE</publisher><subject>balanced packings ; Binary codes ; Binary system ; Codes ; Combinatorial analysis ; Composition ; Constant-composition codes ; difference families ; Frequency modulation ; Indexes ; Johnson-type bound ; Lower bounds ; Memoryless systems ; Nickel ; Upper bound ; Upper bounds</subject><ispartof>IEEE transactions on information theory, 2018-02, Vol.64 (2), p.909-917</ispartof><rights>Copyright The Institute of Electrical and Electronics Engineers, Inc. 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Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula> be the smallest such length. The determination of <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula> is trivial for binary codes. This paper provides a lower bound on <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula>, which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. 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Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula> be the smallest such length. The determination of <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula> is trivial for binary codes. This paper provides a lower bound on <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula>, which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by the refining method, we determine the values of <inline-formula> <tex-math notation="LaTeX">N_{ccc}({\overline {w}}) </tex-math></inline-formula>, for all <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-ary constant-composition codes, provided that <inline-formula> <tex-math notation="LaTeX">3w_{1}\geq w </tex-math></inline-formula> with finite possible exceptions.]]></abstract><cop>New York</cop><pub>IEEE</pub><doi>10.1109/TIT.2017.2689026</doi><tpages>9</tpages><orcidid>https://orcid.org/0000-0001-7823-8068</orcidid><orcidid>https://orcid.org/0000-0003-4234-3610</orcidid></addata></record>
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subjects balanced packings
Binary codes
Binary system
Codes
Combinatorial analysis
Composition
Constant-composition codes
difference families
Frequency modulation
Indexes
Johnson-type bound
Lower bounds
Memoryless systems
Nickel
Upper bound
Upper bounds
title Linear Size Constant-Composition Codes Meeting the Johnson Bound
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