Pregled bibliografske jedinice broj: 1219211
An algorithm for construction of extremal and near- extremal Z4-codes
An algorithm for construction of extremal and near- extremal Z4-codes, 2022., doktorska disertacija, Prirodoslovno-matematički fakultet, Zagreb
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Naslov
An algorithm for construction of extremal and near-
extremal Z4-codes
Autori
Mravić, Matteo
Vrsta, podvrsta i kategorija rada
Ocjenski radovi, doktorska disertacija
Fakultet
Prirodoslovno-matematički fakultet
Mjesto
Zagreb
Datum
01.07
Godina
2022
Stranica
129
Mentor
Rukavina, Sanja
Ključne riječi
Extremal Z_4-code, near-extremal Z_4-code, Z_4-code, binary linear code, self-dual code, doubly-even code, Hadamard matrix, Hadamard design, algorithm, Euclidean weight
Sažetak
The main subjects of this thesis are extremal and near-extremal Z4-codes. The known method of constructing a self-dual Z4-code has a doubly-even binary code of dimension k as a starting point. It consists of choosing one matrix B among the 2^k(k+1)/2 binary k ×k matrices suitable for constructing a self-dual Z4-code. The usual approach to the construction of extremal or near-extremal Z4-codes consists of the construction of self- dual Z4-codes and checking their minimum Euclidean weight. The calculation of the minimum Euclidean weight is necessary to determine the extremality or near-extremality of Z4-codes, but it is also time-consuming. Calculation of the minimum Euclidean weight of the code gets slower as the length of the examined code gets bigger. This fact, together with the size of the search space, makes this method inefficient. In this thesis, we modified the known method in such a way that more than one Z4-code can be checked for extremality or near- extremality, from one calculation of the minimum Euclidean weight. This increases the efficiency of the existing method. Also, we developed a method to construct a series of Hadamard designs on 4n − 1 points from one skew-symmetric Hadamard matrix of order n. It is known that the incidence matrix of a Hadamard 3-design spans a doubly-even binary code. We used developed algorithms to construct new extremal Z4-codes of lengths 32 and 40. On length 48, by our modified method, we obtained already known extremal Z4-code, and at least two nonequivalent near-extremal Z4-codes. From obtained codes of lengths 32 and 40, we constructed strongly regular graphs. All of the obtained graphs were already known. Also, we constructed 1-designs on 32 points from obtained extremal Z4-codes of length 32. Some of the constructed designs are resolvable.
Izvorni jezik
Engleski
Znanstvena područja
Matematika
POVEZANOST RADA
Projekti:
HRZZ-IP-2018-01-6732 - Kombinatorički objekti i kodovi (COCo) (Crnković, Dean, HRZZ ) ( CroRIS)
NadSve-Sveučilište u Rijeci-uniri-prirod-18-45 - Dizajni, grafovi i linearni kodovi (Rukavina, Sanja, NadSve - UNIRI PROJEKTI 2018) ( CroRIS)
Ustanove:
Sveučilište u Rijeci, Fakultet za matematiku