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Course, academic year 2023/2024
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Linear Algebra Applications in Combinatorics - NDMX028
Title: Aplikace lineární algebry v kombinatorice
Guaranteed by: Student Affairs Department (32-STUD)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Is provided by: NDMI028
Guarantor: prof. RNDr. Jan Kratochvíl, CSc.
Class: Informatika Mgr. - Diskrétní modely a algoritmy
M Mgr. MSTR > Povinně volitelné
Classification: Informatics > Discrete Mathematics
Pre-requisite : {NXXX007, NXXX008, NXXX009, NXXX051, NXXX052, NXXX053}
Incompatibility : NDMI028
Interchangeability : NDMI028
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Annotation -
Last update: T_KAM (20.04.2007)
Advanced course in Computer Science Applications of linear algebraic methods in graph theory and combinatorics. Linear dependence and independence of vectors, equiangular lines, two-distance sets, almost disjoint set systems. Determinants. Eigenvalues and eigenvectors, Moore graphs, strongly regular graphs. Seidel's switching. Error-correcting codes, namely perfect codes in Hamming metrics. Theory of distance regular graphs and Biggs's proof of Lloyd's theorem. Van Lint-Tietavainen's proof of nonexistence of perfect codes over finite fields.
Course completion requirements -
Last update: prof. RNDr. Jan Kratochvíl, CSc. (16.09.2020)

Solution of home assignments is required for the credit.

Literature - Czech
Last update: T_KAM (20.04.2007)

Cvetkovic, Doob, Sachs: Spectra of graphs Biggs: Algebraic graph theory

Sloane, McWilliams: Coding theory

Requirements to the exam -
Last update: prof. RNDr. Jan Kratochvíl, CSc. (23.09.2020)

The exam is oral and may be performed remotely. The knowledge and skills examined correspond to the syllabus in extent presented during the lectures. Common understanding to all notions and their relationship, theorems including proofs and ability to apply the acquired skills to simple situations related to the topics of the class are subject of the examination. Credit from the recitations must be obtained prior to enrolling to an exam.

Syllabus -
Last update: prof. RNDr. Jan Kratochvíl, CSc. (18.10.2018)

Application of linear dependence and independence - cardinality of nearly-disjoint set systems, equiangular line systems, two-distance point sets.

Set systems with prescribed parity of intersections.

Eigenvalue techniques - spectra of graphs, interlacing of eigenvalues, Moore graphs.

Perfect codes in Hamming metrics and generalization to distance-regular graphs, Biggs's proof of Lloyd theorem, van Lint-Tietavainen proof of nonexistence of perfect codes over finite fields.

Seidel's switching.

Construction of Golay codes.

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