On the structure of non-full-rank perfect $q$-ary codes

Denis Krotov, Olof Heden
2011 Advances in Mathematics of Communications  
The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the q-ary case. Simply, every non-full-rank perfect code C is the union of a well-defined family of μ-components K_μ, where μ belongs to an "outer" perfect code C^*, and these components are at distance three from each other. Components from distinct
more » ... codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain μ-components, and new lower bounds on the number of perfect 1-error-correcting q-ary codes are presented.
doi:10.3934/amc.2011.5.149 fatcat:mafi3tvyxng6hjpgplk4bins6u