1-popolne kode v kockam podobnih grafih

magistrsko delo

Ana Kompan (Author), Sandi Klavžar (Mentor)

Abstract

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.

Keywords

hiperkocke;dualne kocke;Hammingove kode;1-popolne kode;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[A. Kompan]
UDC:	519.1
COBISS:	18388825
Views:	719
Downloads:	352
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	1-perfect codes over dual-cubes
Secondary abstract:	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.
Secondary keywords:	hypercubes;dual-cubes;Hamming codes;1-perfect codes;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja
Pages:	56 str.
ID:	10941304