On some new extremal Type II Z4-codes of length 40

Using the building-up method and a modification of the doubling method we construct new extremal Type II Z4-codes of length 40. The constructed codes of type $4^{k_1}2^{k_2}$, for $k_1in {8,9,10,11,12,14,15}$, are the first examples of extremal Type II Z4-codes of given type and length 40 whose residue codes have minimum weight greater than or equal to 8. Further, we use minimum weight codewords for a construction of 1-designs, some of which are self-orthogonal

Konstrukcija ekstremalnih Z4-kodova tipa II

Predmet istraživanja ove doktorske disertacije je konstrukcija ekstremalnih $\mathbb{Z}_4$-kodova tipa II, odnosno samodulanih $\mathbb{Z}_4$-kodova kod kojih su euklidske težine svih riječi djeljive sa 8 i koji imaju najveću moguću minimalnu euklidsku težinu. Ekstremalne $\mathbb{Z}_4$-kodove tipa II konstruiramo iz postojećih ekstremalnih $\mathbb{Z}_4$-kodova tipa II i iz binarnih kodova s određenim svojstvima. Cilj je konstrukcija novih ekstremalnih $\mathbb{Z}_4$-kodova tipa II. Pri konstrukcijama, osim teorijskih rezultata, koristimo programe napisane za uporabu u programskim paketima GAP ([40]) i Magma ([6]). Polazeći od poznate metode konstrukcije ekstremalnih $\mathbb{Z}_4$-kodova tipa II za duljine 24; 32 i 40; koja je prikazana u [8], razvijamo metodu konstrukcije ekstremalnih $\mathbb{Z}_4$-kodova tipa II za duljine 48; 56 i 64: Primjenom razvijene metode te otprije poznatih metoda iz [35] i [8], pokušavamo konstruirati nove ekstremalne $\mathbb{Z}_4$-kodove tipa II. Također, ispitujemo svojstva binarnih kodova pridruženih Hadamardovim matricama te njihovu podobnost za korištenje u konstrukciji ekstremalnih $\mathbb{Z}_4$-kodova tipa II. Na ekstremalne $\mathbb{Z}_4$-kodove tipa II dobivene polazeći od Hadamardovih matrica primjenjujemo poznate metode iz [8] te u disertaciji razvijene metode s ciljem dobivanja novih ekstremalnih $\mathbb{Z}_4$-kodova tipa II. Konstruirani su novi ekstremalni $\mathbb{Z}_4$-kodovi tipa II duljina 32 i 40. Iz dobivenih novih ekstremalnih $\mathbb{Z}_4$-kodova tipa II duljine 32 konstruiramo kombinatoričke dizajne i grafove koristeći nosače riječi određenih težina te ispitujemo svojstva dobivenih kombinatoričkih struktura.The main subject of this dissertation is the construction of extremal Type II $\mathbb{Z}_4$-codes. The aim is to construct new extremal Type II $\mathbb{Z}_4$-codes. We constructed extremal Type II $\mathbb{Z}_4$-codes using known methods and methods developed in this dissertation starting from existing extremal Type II $\mathbb{Z}_4$-codes. We also constructed Type II $\mathbb{Z}_4$-codes from certain binary codes and examined the extremality of obtained codes. We have been using GAP and Magma software packages. Some necessary concepts from coding theory, design theory, graph theory and basic facts about lattices are introduced in the first chapter. Second chapter of the dissertation contains the classification of extremal Type II $\mathbb{Z}_4$-codes of lengths 8, 16 and 24 and information about numbers of known extremal Type II $\mathbb{Z}_4$-codes of larger lengths. It also contains constructions of self-dual $\mathbb{Z}_4$-codes and Type II $\mathbb{Z}_4$-codes from binary codes with certain properties. We have been trying to construct new extremal Type II $\mathbb{Z}_4$-codes of lengths 56 and 64 starting from some binary codes. Third chapter of the dissertation contains three constructions of extremal Type II $\mathbb{Z}_4$-codes from existing extremal Type II $\mathbb{Z}_4$-codes. We corrected errors in the first construction and obtained new extremal Type II $\mathbb{Z}_4$-codes of length 32 in that way. Starting from known methods for construction of extremal Type II $\mathbb{Z}_4$-codes of lengths 24; 32 and 40 we developed a method for construction of extremal Type II $\mathbb{Z}_4$-codes of lengths 48; 56 and 64. We have been trying to construct new extremal Type II $\mathbb{Z}_4$-codes using this method. The second construction was used to get new extremal Type II $\mathbb{Z}_4$-codes of length 40. We tried to construct new extremal Type II $\mathbb{Z}_4$-codes of lengths 56 and 64 starting from certain existing extremal Type II $\mathbb{Z}_4$-codes of lengths 56 and 64 using the first and the third construction in this chapter. We showed that new extremal Type II $\mathbb{Z}_4$-codes of lengths 56 and 64 cannot be obtained in this way. In the next part of the dissertation we looked at the binary codes associated with Hadamard matrices. Their properties were analyzed. We proved their suitability for construction of extremal Type II $\mathbb{Z}_4$-codes introduced in the second chapter of the dissertation. New extremal Type II $\mathbb{Z}_4$-codes of lengths 32 and 40 were obtained with the use of Hadamard matrices. The last, fifth chapter, is devoted to constructing combinatorial designs and graphs using supports of codewords with specific weights of the obtained new extremal Type II $\mathbb{Z}_4$-codes of length 32. Properties of the constructed structures were analyzed. We tried to construct new designs and distance-regular or strongly-regular graphs. Parameters of all obtained designs are known. We obtained some regular graphs which are not distance-regular

Konstrukcija ekstremalnih Z4-kodova tipa II

Predmet istraživanja ove doktorske disertacije je konstrukcija ekstremalnih $\mathbb{Z}_4$-kodova tipa II, odnosno samodulanih $\mathbb{Z}_4$-kodova kod kojih su euklidske težine svih riječi djeljive sa 8 i koji imaju najveću moguću minimalnu euklidsku težinu. Ekstremalne $\mathbb{Z}_4$-kodove tipa II konstruiramo iz postojećih ekstremalnih $\mathbb{Z}_4$-kodova tipa II i iz binarnih kodova s određenim svojstvima. Cilj je konstrukcija novih ekstremalnih $\mathbb{Z}_4$-kodova tipa II. Pri konstrukcijama, osim teorijskih rezultata, koristimo programe napisane za uporabu u programskim paketima GAP ([40]) i Magma ([6]). Polazeći od poznate metode konstrukcije ekstremalnih $\mathbb{Z}_4$-kodova tipa II za duljine 24; 32 i 40; koja je prikazana u [8], razvijamo metodu konstrukcije ekstremalnih $\mathbb{Z}_4$-kodova tipa II za duljine 48; 56 i 64: Primjenom razvijene metode te otprije poznatih metoda iz [35] i [8], pokušavamo konstruirati nove ekstremalne $\mathbb{Z}_4$-kodove tipa II. Također, ispitujemo svojstva binarnih kodova pridruženih Hadamardovim matricama te njihovu podobnost za korištenje u konstrukciji ekstremalnih $\mathbb{Z}_4$-kodova tipa II. Na ekstremalne $\mathbb{Z}_4$-kodove tipa II dobivene polazeći od Hadamardovih matrica primjenjujemo poznate metode iz [8] te u disertaciji razvijene metode s ciljem dobivanja novih ekstremalnih $\mathbb{Z}_4$-kodova tipa II. Konstruirani su novi ekstremalni $\mathbb{Z}_4$-kodovi tipa II duljina 32 i 40. Iz dobivenih novih ekstremalnih $\mathbb{Z}_4$-kodova tipa II duljine 32 konstruiramo kombinatoričke dizajne i grafove koristeći nosače riječi određenih težina te ispitujemo svojstva dobivenih kombinatoričkih struktura.The main subject of this dissertation is the construction of extremal Type II $\mathbb{Z}_4$-codes. The aim is to construct new extremal Type II $\mathbb{Z}_4$-codes. We constructed extremal Type II $\mathbb{Z}_4$-codes using known methods and methods developed in this dissertation starting from existing extremal Type II $\mathbb{Z}_4$-codes. We also constructed Type II $\mathbb{Z}_4$-codes from certain binary codes and examined the extremality of obtained codes. We have been using GAP and Magma software packages. Some necessary concepts from coding theory, design theory, graph theory and basic facts about lattices are introduced in the first chapter. Second chapter of the dissertation contains the classification of extremal Type II $\mathbb{Z}_4$-codes of lengths 8, 16 and 24 and information about numbers of known extremal Type II $\mathbb{Z}_4$-codes of larger lengths. It also contains constructions of self-dual $\mathbb{Z}_4$-codes and Type II $\mathbb{Z}_4$-codes from binary codes with certain properties. We have been trying to construct new extremal Type II $\mathbb{Z}_4$-codes of lengths 56 and 64 starting from some binary codes. Third chapter of the dissertation contains three constructions of extremal Type II $\mathbb{Z}_4$-codes from existing extremal Type II $\mathbb{Z}_4$-codes. We corrected errors in the first construction and obtained new extremal Type II $\mathbb{Z}_4$-codes of length 32 in that way. Starting from known methods for construction of extremal Type II $\mathbb{Z}_4$-codes of lengths 24; 32 and 40 we developed a method for construction of extremal Type II $\mathbb{Z}_4$-codes of lengths 48; 56 and 64. We have been trying to construct new extremal Type II $\mathbb{Z}_4$-codes using this method. The second construction was used to get new extremal Type II $\mathbb{Z}_4$-codes of length 40. We tried to construct new extremal Type II $\mathbb{Z}_4$-codes of lengths 56 and 64 starting from certain existing extremal Type II $\mathbb{Z}_4$-codes of lengths 56 and 64 using the first and the third construction in this chapter. We showed that new extremal Type II $\mathbb{Z}_4$-codes of lengths 56 and 64 cannot be obtained in this way. In the next part of the dissertation we looked at the binary codes associated with Hadamard matrices. Their properties were analyzed. We proved their suitability for construction of extremal Type II $\mathbb{Z}_4$-codes introduced in the second chapter of the dissertation. New extremal Type II $\mathbb{Z}_4$-codes of lengths 32 and 40 were obtained with the use of Hadamard matrices. The last, fifth chapter, is devoted to constructing combinatorial designs and graphs using supports of codewords with specific weights of the obtained new extremal Type II $\mathbb{Z}_4$-codes of length 32. Properties of the constructed structures were analyzed. We tried to construct new designs and distance-regular or strongly-regular graphs. Parameters of all obtained designs are known. We obtained some regular graphs which are not distance-regular