Perfect binary codes: classification and properties

An r-perfect binary code is a subset of ℤ2n 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 2m−42^m-4 and 2m−32^m-3

Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length $2^m-4$ and $2^m-3$, 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 $m \geq 4$, there are optimal binary one-error-correcting codes of length $2^m-4$ and $2^m-3$ that cannot be lengthened to perfect codes of length $2^m-1$.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. Kyseessä on jo jonkin aikaa avoimena olleen laskennallisen ongelman ratkaiseminen. Työn alussa tarkastellaan Steinerin systeemeitä teoreettiselta kannalta, kuitenkin luokitteluun liittyviin tuloksiin keskittyen. Nelikkosysteemien olemassaoloa ja lukumäärää tutkitaan, kuten myös niiden yhteyttä Steinerin kolmikkosysteemeihin ja tiettyihin koodeihin. Myös Pasch-konfiguraatioita ja niiden hyödyntämistä isomorfiatarkasteluissa tarkastellaan. McKayn kehittämä luokittelumenetelmä, kanonisilla lisäyksillä tuottaminen, esitellään varsin yleisellä tasolla. Menetelmää soveltamalla kehitetään luokittelualgoritmi Steinerin nelikkosysteemeille. Lisäksi esitetään Kasken ja Östergårdin kehittämä samankaltainen algoritmi Steinerin kolmikkosysteemeille. Myös vaihtoehtoinen, Zinovievin ja Zinovievin kehittämä luokittelumenetelmä esitellään lyhyesti. Nelikkosysteemejä tuotettaessa ja isomorfiakarsintaa suoritettaessa kohdataan kaksi vaikeaa osaongelmaa: täsmällisten peitteiden etsiminen tietyille joukoille ja systeemeiden kanonisen nimeämisen laskeminen. Näitä ongelmia ja niiden vaativuutta tarkastellaan. Vaikka käytettävän algoritmin oikeellisuus on todistettu matemaattisella tarkkuudella, voi ohjelmointivirhe johtaa virheellisiin tuloksiin. Tällaisten mahdollisten virheiden havaitsemiseksi testattiin laskennan tulosten johdonmukaisuutta. Luokittelun tuloksena saatiin yksi edustaja jokaisesta 16 pisteen Steinerin nelikkosysteemien isomorfialuokasta. Isomorfialuokkia on yhteensä 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 ≥ 4