On the structure of non-full-rank perfect $q$-ary codes
Olof Heden Denis S. Krotov
Advances in Mathematics of Communications 2011, 5(2): 149-156 doi: 10.3934/amc.2011.5.149
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 speaking, every non-full-rank perfect code $C$ is the union of a well-defined family of $\bar\mu$-components K$\bar\mu$, where $\bar\mu$ belongs to an “outer” perfect code C*, and these components are at distance three from each other. Components from distinct 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 $\bar\mu$-components, and new lower bounds on the number of perfect $1$-error-correcting $q$-ary codes are presented.
keywords: lower bound. $q$-ary codes Perfect codes components
Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code
Denis S. Krotov Patric R. J.  Östergård Olli Pottonen
Advances in Mathematics of Communications 2016, 10(2): 393-399 doi: 10.3934/amc.2016013
Ternary constant weight codes of length $n=2^m$, weight $n-1$, cardinality $2^n$ and distance $5$ are known to exist for every $m$ for which there exists an APN permutation of order $2^m$, that is, at least for all odd $m \geq 3$ and for $m=6$. We show the non-existence of such codes for $m=4$ and prove that any codes with the parameters above are diameter perfect.
keywords: diameter perfect code. Constant weight code

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