33,359 research outputs found

    On the number of nonequivalent propelinear extended perfect codes

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    The paper proves that there exist an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the properties of these codes and show that any of them has a normalized propelinear representation

    On the structure of non-full-rank perfect codes

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    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 qq-ary case. Simply, every non-full-rank perfect code CC is the union of a well-defined family of μ\mu-components KμK_\mu, where μ\mu belongs to an "outer" perfect code C∗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 μ\mu-components, and new lower bounds on the number of perfect 1-error-correcting qq-ary codes are presented.Comment: 8 page

    On the number of 1-perfect binary codes: a lower bound

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    We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.Comment: 5pp(Eng)+7pp(Rus) V2: revised V3: + Russian version, + reference
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