20.500.12556/RUL-101536 1-popolne kode v kockam podobnih grafih magistrsko delo 1-perfect codes over dual-cubes V magistrskem delu spoznamo kodiranje, kode in popolne kode, njihovo posplošitev na grafe ter povezavo med 1-popolnimi kodami in dominantno množico v grafih. Opišemo hiperkocke in dokažemo nekaj njihovih lastnosti. Spoznamo grafe dualnih kock, ki so sorodni hiperkockam. Pokažemo, katere lastnosti dualna kocka podeduje od ustrezne hiperkocke, in tiste, ki so pomembne za tvorjenje 1-popolne kode. Ugotovimo, da je dualna kocka $DQ_m$ sestavljena iz $2^{m+1}$ induciranih hiperkock. Le-te namreč vsebujejo Hammingove kode, ki so 1-popolne kode in to vzamemo kot osnovo za tvorjenje 1-popolne kode v dualni kocki. Nato Hammingove kode z nadaljnjimi algoritmi preoblikujemo tako, da res nastane 1-popolna koda v dualni kocki. S tem dokažemo izrek, da dualna kocka $DQ_m$ vsebuje 1-popolno kodo natanko tedaj, ko je $m=2^k-2$ za $k\ge 2$. Dobljeni rezultat uporabimo pri postavljanju meja za dominanto število v dualni kocki s poljubnim parametrom. In this thesis we introduce coding, codes and perfect codes, their generalization to graphs, and a connection between 1-perfect codes and dominating sets in graphs. We describe hypercubes and prove some of their characteristics. Dual-cubes, which are similar to hypercubes, are introduced. We show the characteristics that are inherited from hypercubes and some that are important for generating 1-perfect codes. We prove that the dual-cube $DQ_m$ consists of $2^{m+1}$ induced hypercubes. Hypercubes contain Hamming codes which are 1-perfect codes, and this is taken as a basis of creating 1-perfect codes in dual-cubes. Hamming codes are further transformed with algorithms that eventually lead to 1-perfect codes. With this we show that the dual-cube $DQ_m$ admits a 1-perfect code if and only if $m=2^k-2$ for $k\ge 2$. This result is used for proving tight bounds on the domination number of dual-cube with an arbitrary parameter. hiperkocke dualne kocke Hammingove kode 1-popolne kode hypercubes dual-cubes Hamming codes 1-perfect codes true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2018-06-14 07:46:06 2018-06-14 07:46:07 2022-08-14 03:44:59 0000-00-00 00:00:00 2018 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.1 87337 18388825 76.pdf 76.pdf 1 835BDD89E11C62BC65F6C2E52EB53BEA 9401c5eda4d08ccdea44bdb11384ef2aad4fc94d0247ae5f8f2bd434af899446 4d270b21-a1b5-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=111609 Fakulteta za matematiko in fiziko 0 0 0