22 research outputs found
Perfect binary codes: classification and properties
An r-perfect binary code is a subset of â€2n such that for any word, there is a unique codeword at Hamming distance at most r. Such a code is r-error-correcting. Two codes are equivalent if one can be obtained from the other by permuting the coordinates and adding a constant vector. The main result of this thesis is a computer-aided classification, up to equivalence, of the 1-perfect binary codes of length 15.
In an extended 1-perfect code, the neighborhood of a codeword corresponds to a Steiner quadruple system. To utilize this connection, we start with a computational classification of Steiner quadruple systems of order 16. This classification is also used to establish the nonexistence of Steiner quintuple systems S(4, 5, 17).
The classification of the codes is used for computational examination of their properties. These properties include occurrences of Steiner triple and quadruple systems, automorphisms, ranks, structure of i-components and connections to orthogonal arrays and mixed perfect codes.
It is also proved that extended 1-perfect binary codes are equivalent if and only if their minimum distance graphs are isomorphic
The Perfect Binary One-Error-Correcting Codes of Length 15: Part II--Properties
A complete classification of the perfect binary one-error-correcting codes of
length 15 as well as their extensions of length 16 was recently carried out in
[P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary
one-error-correcting codes of length 15: Part I--Classification," IEEE Trans.
Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying
work, the classified codes are studied in great detail, and their main
properties are tabulated. The results include the fact that 33 of the 80
Steiner triple systems of order 15 occur in such codes. Further understanding
is gained on full-rank codes via switching, as it turns out that all but two
full-rank codes can be obtained through a series of such transformations from
the Hamming code. Other topics studied include (non)systematic codes, embedded
one-error-correcting codes, and defining sets of codes. A classification of
certain mixed perfect codes is also obtained.Comment: v2: fixed two errors (extension of nonsystematic codes, table of
coordinates fixed by symmetries of codes), added and extended many other
result
On Optimal Binary One-Error-Correcting Codes of Lengths and
Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that
triply-shortened and doubly-shortened binary Hamming codes (which have length
and , respectively) are optimal. Properties of such codes are
here studied, determining among other things parameters of certain subcodes. A
utilization of these properties makes a computer-aided classification of the
optimal binary one-error-correcting codes of lengths 12 and 13 possible; there
are 237610 and 117823 such codes, respectively (with 27375 and 17513
inequivalent extensions). This completes the classification of optimal binary
one-error-correcting codes for all lengths up to 15. Some properties of the
classified codes are further investigated. Finally, it is proved that for any
, there are optimal binary one-error-correcting codes of length
and that cannot be lengthened to perfect codes of length
.Comment: Accepted for publication in IEEE Transactions on Information Theory.
Data available at http://www.iki.fi/opottone/code
The Perfect Binary One-Error-Correcting Codes of Length 15: Part I--Classification
A complete classification of the perfect binary one-error-correcting codes of
length 15 as well as their extensions of length 16 is presented. There are 5983
such inequivalent perfect codes and 2165 extended perfect codes. Efficient
generation of these codes relies on the recent classification of Steiner
quadruple systems of order 16. Utilizing a result of Blackmore, the optimal
binary one-error-correcting codes of length 14 and the (15, 1024, 4) codes are
also classified; there are 38408 and 5983 such codes, respectively.Comment: 6 pages. v3: made the codes available in the source of this pape
Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs
The minimum distance graph of a code has the codewords as vertices and edges
exactly when the Hamming distance between two codewords equals the minimum
distance of the code. A constructive proof for reconstructibility of an
extended perfect binary one-error-correcting code from its minimum distance
graph is presented. Consequently, inequivalent such codes have nonisomorphic
minimum distance graphs. Moreover, it is shown that the automorphism group of a
minimum distance graph is isomorphic to that of the corresponding code.Comment: 4 pages. Accepted for publication in IEEE Transactions on Information
Theor
Two Optimal One-Error-Correcting Codes of Length 13 That Are Not Doubly Shortened Perfect Codes
The doubly shortened perfect codes of length 13 are classified utilizing the
classification of perfect codes in [P.R.J. \"Osterg{\aa}rd and O. Pottonen, The
perfect binary one-error-correcting codes of length 15: Part I -
Classification, IEEE Trans. Inform. Theory, to appear]; there are 117821 such
(13,512,3) codes. By applying a switching operation to those codes, two more
(13,512,3) codes are obtained, which are then not doubly shortened perfect
codes.Comment: v2: a correction concerning shortened codes of length 1
Steinerin nelikkosysteemien luokittelu
Työn tavoitteena on luokitella 16 pisteen Steinerin nelikkosysteemit.
KyseessÀ on jo jonkin aikaa avoimena olleen laskennallisen ongelman ratkaiseminen.
Työn alussa tarkastellaan Steinerin systeemeitÀ teoreettiselta kannalta, kuitenkin luokitteluun liittyviin tuloksiin keskittyen.
Nelikkosysteemien olemassaoloa ja lukumÀÀrÀÀ tutkitaan, kuten myös niiden yhteyttÀ Steinerin kolmikkosysteemeihin ja tiettyihin koodeihin.
Myös Pasch-konfiguraatioita ja niiden hyödyntÀmistÀ isomorfiatarkasteluissa tarkastellaan.
McKayn kehittÀmÀ luokittelumenetelmÀ, kanonisilla lisÀyksillÀ tuottaminen, esitellÀÀn varsin yleisellÀ tasolla.
MenetelmÀÀ soveltamalla kehitetÀÀn luokittelualgoritmi Steinerin nelikkosysteemeille.
LisĂ€ksi esitetÀÀn Kasken ja ĂstergĂ„rdin kehittĂ€mĂ€ samankaltainen algoritmi Steinerin kolmikkosysteemeille.
Myös vaihtoehtoinen, Zinovievin ja Zinovievin kehittÀmÀ luokittelumenetelmÀ esitellÀÀn lyhyesti.
NelikkosysteemejÀ tuotettaessa ja isomorfiakarsintaa suoritettaessa kohdataan kaksi vaikeaa osaongelmaa: tÀsmÀllisten peitteiden etsiminen tietyille joukoille ja systeemeiden kanonisen nimeÀmisen laskeminen.
NÀitÀ ongelmia ja niiden vaativuutta tarkastellaan.
Vaikka kÀytettÀvÀn algoritmin oikeellisuus on todistettu matemaattisella tarkkuudella, voi ohjelmointivirhe johtaa virheellisiin tuloksiin.
TĂ€llaisten mahdollisten virheiden havaitsemiseksi testattiin laskennan tulosten johdonmukaisuutta.
Luokittelun tuloksena saatiin yksi edustaja jokaisesta 16 pisteen Steinerin nelikkosysteemien isomorfialuokasta.
Isomorfialuokkia on yhteensÀ 1,054,163 kappaletta.
Luokkien edustajia tutkimalla saatiin selville joitain uusia tuloksia, kuten resolvoitumattoman 16 pisteen Steinerin nelikkosysteemin olemassaolo
There exists no Steiner system S (4, 5, 17)
If a Steiner system S (4, 5, 17) exists, it would contain derived S (3, 4, 16) designs. By relying on a recent classification of the S (3, 4, 16), an exhaustive computer search for S (4, 5, 17) is carried out. The search shows that no S (4, 5, 17) exists, thereby ruling out the existence of Steiner systems S (t, t + 1, t + 13) for t â„ 4