25 research outputs found
On the structure of non-full-rank perfect codes
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 -ary case. Simply, every non-full-rank
perfect code is the union of a well-defined family of -components
, where belongs to an "outer" perfect code , 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 -components, and new lower bounds on the number of perfect
1-error-correcting -ary codes are presented.Comment: 8 page
Around the Hossz\'u-Gluskin theorem for -ary groups
We survey results related to the important Hossz\'u-Gluskin Theorem on
-ary groups adding also several new results and comments. The aim of this
paper is to write all such results in uniform and compressive forms. Therefore
some proofs of new results are only sketched or omitted if their completing
seems to be not too difficult for readers. In particular, we show as the
Hossz\'u-Gluskin Theorem can be used for evaluation how many different -ary
groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the
mentioned theorem can be also used for investigation of
-independent subsets of semiabelian -ary groups for some
special families of mappings
T-functions revisited: New criteria for bijectivity/transitivity
The paper presents new criteria for bijectivity/transitivity of T-functions
and fast knapsack-like algorithm of evaluation of a T-function. Our approach is
based on non-Archimedean ergodic theory: Both the criteria and algorithm use
van der Put series to represent 1-Lipschitz -adic functions and to study
measure-preservation/ergodicity of these
Wild chimeras: Enthusiasm and intellectual virtue in Kant
Kant typically is not identified with the tradition of virtue epistemology. Although he may not be a virtue epistemologist in a strict sense, I suggest that intellectual virtues and vices play a key role in his epistemology. Specifically, Kant identifies a serious intellectual vice that threatens to undermine reason, namely enthusiasm (SchwĂ€rmerei). Enthusiasts become so enamored with their own thinking that they refuse to subject reason to selfâcritique. The particular danger of enthusiasm is that reason colludes in its own destruction: Enthusiasm occurs when selfâconceit and reason\u27s desire to transcend its boundaries mutually reinforce each other. I conclude by sketching an account of Kantian intellectual virtue that is consistent with Kantian moral virtue