Self-dual codes and PD-sets constructed from combinatorial designs

Abstract

Predmet istraživanja ove doktorske disertacije su kodovi konstruirani iz nekih kombinatoričkih dizajna i njihova svojstva. Uvodno, u prvom su poglavlju izloženi pojmovi iz teorije grupa potrebni u nastavku, te osnove teorije kodiranja, grafova i dizajna. Zatim su u drugom poglavlju disertacije promatrani kodovi razapeti retcima kvocijentne matrice simetričnog (grupovno) djeljivog dizajna (SGDD) s dualnim svojstvom. Definirana je proširena kvocijentna matrica i pokazano je da pod određenim uvjetima retci proširene kvocijentne matrice razapinju samodualan kod u odnosu na određeni skalarni produkt. Također je pokazano da se ponekad lanac kodova može koristiti da pridružimo samodualan kod kvocijentnoj matrici SGDD-a s dualnim svojstvom. Navedeni su rezultati objavljeni u članku [15] čiji su autori Crnković, Mostarac i Rukavina. Tamo su razvijane ideje koje su prezentirali Lander [35] i Wilson [52], te posebno one iz [17], gdje su Crnković i Rukavina dali konstrukciju samodualnih kodova iz proširenih orbitnih matrica simetričnih dizajna. Zatim su opisani i primjeri samodualnih kodova dobivenih opisanom konstrukcijom uz pomoć grafova i digrafova-djeljivih dizajna. Treće poglavlje sadrži konstrukcije samoortogonalnih i samodualnih kodova iz proširenih orbitnih matrica blokovnih dizajna. U njemu su također opisane i konstrukcije samodualnih kodova uz pomoć orbitnih matrica simetričnih dizajna, te analogne konstrukcije pomoću kvocijentnih matrica SGDD-a s dualnim svojstvom, pri čemu su ideje za njih proizašle iz teorema Assmusa, Mezzarobe i Salwacha u [2]. Kao specijalan slučaj jedne od konstrukcija dana je i konstrukcija uz pomoć Hadamardovih dizajna. Opisano je i kako nam Kroneckerov produkt može pomoći u dobivanju samodualnih kodova. Četvrto je poglavlje posvećeno pronalaženju PD-skupova iz flag-tranzitivnih simetričnih dizajna. Za prost broj $p$ neka je $C_p(G)$ $p$-narni kod razapet retcima matrice incidencije $G$ grafa $\Gamma$. Neka je $\Gamma$ incidencijski graf flag-tranzitivnog simetričnog dizajna $\mathcal{D}$. Pokazano je da se bilo koja flag-tranzitivna grupa automorfizama od $\mathcal{D}$ može koristiti kao PD-skup za potpuno ispravljanje pogrešaka za linearan kod $C_p(G)$ (za bilo koji informacijski skup). Dakle, tako dobiveni kodovi mogu se permutacijski dekodirati. Rezultat je poopćen i za kodove iz flag-tranzitivnih SGGD-a s dualnim svojstvom. PD-skupovi dobiveni na opisani način obično su velike kardinalnosti, no proučavanjem primjera kodova proizašlih iz nekih flag-tranzitivnih simetričnih dizajna pokazali smo da se za njih mogu naći manji PD-skupovi za specifične informacijske skupove.The main subject of this thesis are codes constructed from certain combinatorial designs and their properties. We have constructed some self-dual codes obtained with the use of symmetric (group) divisible designs with the dual property. Self-dual codes obtained with the use of block designs have also been constructed. Next, we have shown that codes spanned by the rows of the incidence matrix of the incidence graph of a flag-transitive symmetric design, are permutation decodable. Some necessary concepts from group theory, and also basic concepts from coding theory, graph theory and design theory are introduced in the first chapter. In the second chapter of the thesis we looked at codes spanned by the rows of a quotient matrix of a symmetric (group) divisible design with the dual property. We defined an extended quotient matrix and showed that under certain conditions the rows of the extended quotient matrix span a self-dual code with respect to a certain scalar product. We also showed that sometimes a chain of codes can be used to associate a self-dual code to a quotient matrix of a symmetric group divisible design with the dual property. This was published in the article [15] whose authors are Crnković, Mostarac and Rukavina. There we developed ideas presented by Lander [35] and Wilson [52], and especially from [17], where Crnković and Rukavina gave a construction of self-dual codes from extended orbit matrices of symmetric designs. Then some examples of self-dual codes are given, that were obtained on the described way, using divisible design graphs and divisible design digraphs. The next part of the thesis contains constructions of self-orthogonal and self-dual codes from extended orbit matrices of block designs. It also contains constructions of self-dual codes obtained with the use of orbit matrices of symmetric designs, and analog constructions obtained with the use of quotient matrices of symmetric group divisible designs with the dual property, ideas for which were taken from a theorem of Assmus, Mezzaroba and Salwach in [2]. As a special case of one of the constructions we describe a construction from orbit matrices of Hadamard designs. We also remark how Kronecker product of matrices can help to obtain some new self-dual codes from the previously constructed ones. The last, fourth chapter, is devoted to finding PD-sets from flag-transitive symmetric designs. For any prime $p$ let $C_p(G)$ be the $p$-ary code spanned by the rows of the incidence 85 matrix $G$ of a graph $\Gamma$. Let $\Gamma$ be the incidence graph of a flag-transitive symmetric design $\mathcal{D}$. We showed that any flag-transitive automorphism group of $\mathcal{D}$ can be used as a PD-set for full error correction for the linear code $C_p(G)$ (with any information set). Therefore, such codes derived from flag-transitive symmetric designs can be decoded using permutation decoding. We noticed that PD-sets obtained in the described way are usually of large cardinality, but by studying some examples of codes arising from flag-transitive symmetric designs we showed that smaller PD-sets can be found for them for specific information sets. The result is also generalized for codes obtained from flag-transitive symmetric group divisible designs with the dual property