by To Tze-ming. / Bibliography: leaves 92-93 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1985

by Ho Kin Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 68-69). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Introduction to Coding Theory --- p.9 / Chapter 1.1 --- Definition of a code --- p.10 / Chapter 1.2 --- Maximum Likelihood Decoding --- p.11 / Chapter 1.3 --- Syndrome Decoding --- p.12 / Chapter 1.4 --- Two Kinds of Errors and Concatenated Code --- p.14 / Chapter 2 --- Basic Knowledge of Algebraic Curve --- p.16 / Chapter 2.1 --- Affine and Projective Curve --- p.16 / Chapter 2.2 --- Regular Functions and Maps --- p.17 / Chapter 2.3 --- Divisors and Differential forms --- p.19 / Chapter 2.4 --- Riemann-Roch Theorem --- p.21 / Chapter 3 --- Construction of Algebraic Geometric Code --- p.23 / Chapter 3.1 --- L-construction --- p.23 / Chapter 3.2 --- Ω -construction --- p.24 / Chapter 3.3 --- Duality --- p.26 / Chapter 4 --- Basic Error Processing --- p.28 / Chapter 4.1 --- Error Locators and Syndromes --- p.28 / Chapter 4.2 --- Finding an Error Locator --- p.29 / Chapter 5 --- Full Error Processing for Code on Curve of Genus1 --- p.34 / Chapter 5.1 --- Syndrome table --- p.34 / Chapter 5.2 --- Syndrome table --- p.36 / Chapter 5.3 --- The algorithm of Full Error Processing --- p.38 / Chapter 5.4 --- A simple Example --- p.40 / Chapter 6 --- General Full Error Processing --- p.47 / Chapter 6.1 --- Row Candidate and Column Candidate --- p.47 / Chapter 6.2 --- Consistency --- p.49 / Chapter 6.3 --- Majority voting --- p.50 / Chapter 6.4 --- Example --- p.53 / Chapter 7 --- Application of Algebraic Geometric Code --- p.60 / Chapter 7.1 --- Communication --- p.60 / Chapter 7.2 --- Cryptosystem --- p.62 / Bibliography

by Sheng Mao. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 49-51). / Abstracts in English and Chinese. / Chapter 1 --- What Is K3 Surface? --- p.2 / Chapter 1.1 --- Algebraic K3 Surface --- p.2 / Chapter 1.1.1 --- Definition and Examples --- p.2 / Chapter 1.1.2 --- Topological Invariants of K3 Surfaces --- p.7 / Chapter 1.2 --- Local Torelli Theorem for K3 surfaces --- p.11 / Chapter 1.3 --- Moduli for polarlized K3 surfaces --- p.17 / Chapter 2 --- Arakelov-Yau Type Inequalities For K3 Surfaces --- p.22 / Chapter 2.1 --- A Short Introduction to Hodge Theory --- p.22 / Chapter 2.1.1 --- Variation of Hodge Structrue --- p.22 / Chapter 2.1.2 --- Degeneration of Variation of Hodge Structure --- p.33 / Chapter 2.2 --- Arakelov-Yau Type Inequalities for Family of K3 Sur- faces Over Curve --- p.40 / Chapter 2.3 --- Application to Rigidity Theorem --- p.44 / Bibliography --- p.49

by Yan Cho Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 122-124). / Abstract also in Chinese. / Chapter 1 --- Complex algebraic curves --- p.6 / Chapter 1.1 --- Foundations --- p.6 / Chapter 1.1.1 --- Hilbert Nullstellensatz --- p.6 / Chapter 1.1.2 --- Complex algebraic curves in C2 --- p.9 / Chapter 1.1.3 --- Complex projective curves in P2 --- p.11 / Chapter 1.1.4 --- Affine and projective curves --- p.13 / Chapter 1.2 --- Algebraic properties of complex projective curves in P2 --- p.16 / Chapter 1.2.1 --- Intersection multiplicity --- p.16 / Chapter 1.2.2 --- Bezout's theorem and its applications --- p.18 / Chapter 1.2.3 --- Cubic curves --- p.21 / Chapter 1.3 --- Topological properties of complex projective curves in P2 --- p.23 / Chapter 1.4 --- Riemann surfaces --- p.26 / Chapter 1.4.1 --- Weierstrass &-function --- p.26 / Chapter 1.4.2 --- Riemann surfaces and examples --- p.27 / Chapter 1.5 --- Differentials on Riemann surfaces --- p.28 / Chapter 1.5.1 --- Holomorphic differentials --- p.28 / Chapter 1.5.2 --- Abel's Theorem for tori --- p.31 / Chapter 1.5.3 --- The Riemann-Roch theorem --- p.32 / Chapter 1.6 --- Singular curves --- p.36 / Chapter 1.6.1 --- Resolution of singularities --- p.37 / Chapter 1.6.2 --- The topology of singular curves --- p.45 / Chapter 2 --- Coding theory --- p.48 / Chapter 2.1 --- An introduction to codes --- p.48 / Chapter 2.1.1 --- Efficient noiseless coding --- p.51 / Chapter 2.1.2 --- The main coding theory problem --- p.56 / Chapter 2.2 --- Linear codes --- p.58 / Chapter 2.2.1 --- Syndrome decoding --- p.63 / Chapter 2.2.2 --- Equivalence of codes --- p.65 / Chapter 2.2.3 --- An introduction to cyclic codes --- p.67 / Chapter 2.3 --- Special linear codes --- p.71 / Chapter 2.3.1 --- Hamming codes --- p.71 / Chapter 2.3.2 --- Simplex codes --- p.72 / Chapter 2.3.3 --- Reed-Muller codes --- p.73 / Chapter 2.3.4 --- BCH codes --- p.75 / Chapter 2.4 --- Bounds on codes --- p.77 / Chapter 2.4.1 --- Spheres in Zn --- p.77 / Chapter 2.4.2 --- Perfect codes --- p.78 / Chapter 2.4.3 --- Famous numbers Ar (n，d) and the sphere-covering and sphere packing bounds --- p.79 / Chapter 2.4.4 --- The Singleton and Plotkin bounds --- p.81 / Chapter 2.4.5 --- The Gilbert-Varshamov bound --- p.83 / Chapter 3 --- Algebraic curves over finite fields and the Goppa codes --- p.85 / Chapter 3.1 --- Algebraic curves over finite fields --- p.85 / Chapter 3.1.1 --- Affine varieties --- p.85 / Chapter 3.1.2 --- Projective varieties --- p.37 / Chapter 3.1.3 --- Morphisms --- p.89 / Chapter 3.1.4 --- Rational maps --- p.91 / Chapter 3.1.5 --- Non-singular varieties --- p.92 / Chapter 3.1.6 --- Smooth models of algebraic curves --- p.93 / Chapter 3.2 --- Goppa codes --- p.96 / Chapter 3.2.1 --- Elementary Goppa codes --- p.96 / Chapter 3.2.2 --- The affine and projective lines --- p.98 / Chapter 3.2.3 --- Goppa codes on the projective line --- p.102 / Chapter 3.2.4 --- Differentials and divisors --- p.105 / Chapter 3.2.5 --- Algebraic geometric codes --- p.112 / Chapter 3.2.6 --- Codes with better rates than the Varshamov- Gilbert bound and calculation of parameters --- p.116 / Bibliography

Wong, Sen. / Thesis submitted in: November 2006. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaf 85). / Abstracts in English and Chinese. / Chapter 1 --- From algebraic sets to Preschemes --- p.5 / Chapter 1.1 --- Affine varieties and prevarieties --- p.5 / Chapter 1.2 --- Affine schemes and preschemes --- p.13 / Chapter 1.3 --- Morphisms --- p.16 / Chapter 1.4 --- Prevarieties vs Preschemes --- p.29 / Chapter 1.5 --- Summary and Products --- p.37 / Chapter 2 --- More on prevarieties --- p.43 / Chapter 2.1 --- Introduction --- p.43 / Chapter 2.2 --- Subprevarieties and product of prevarieties --- p.48 / Chapter 2.3 --- Dominating and birational morphisms --- p.60 / Chapter 2.4 --- Complete varieties --- p.64 / Chapter 2.5 --- Chow's Theorem --- p.68 / Chapter 3 --- From preschemes to schemes --- p.71 / Chapter 3.1 --- Introduction --- p.71 / Chapter 3.2 --- Open subpreschemes and Open immersions --- p.74 / Chapter 3.3 --- Closed subpreschemes and Closed immersions --- p.76 / Chapter 3.4 --- Schemes --- p.79 / Chapter A --- Hilbert's Nullstellensatz --- p.81 / Bibliography --- p.85

The Periodicity theorem of Hopkins and Smith tells us that any finite spectrum supports a $v_n$-map for some $n$. We are interested in finding finite $2$-local spectra that both support a $v_2$-map with a low power of $v_2$ and have few cells. Following the process outlined in Palmieri-Sadofsky, we study a related class of self-maps, known as $u_2$-maps, between stably finite spectra. We construct examples of spectra that might be expected to support $u_2^1$-maps, and then we use Margolis homology and homological algebra computations to show that they do not support $u_2^1$-maps. We also show that one example does not support a $u_2^2$-map. The nonexistence of $u_2$-maps on these spectra eliminates certain examples from consideration by this technique.

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Zhang Jia-jin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Configurations Of Points In P1 And Local Systems Of Rank One --- p.6 / Chapter 2.1 --- Configurations Of Points In P1 --- p.6 / Chapter 2.2 --- Local Systems Of Rank One --- p.7 / Chapter 2.3 --- Arithmeticity And Integral Monodromy --- p.12 / Chapter 3 --- Generalized Jacobians --- p.13 / Chapter 4 --- Stable Reductions Of Family Of Curves --- p.17 / Chapter 4.1 --- Normalization Of Cyclic Branched coverings --- p.17 / Chapter 4.2 --- Stable Reductions --- p.19 / Chapter 5 --- "Family Of n-th Cyclic Coverings Of P1, Abelian Va- rieties And CM-type" --- p.22 / Chapter 5.1 --- Family Of n-th Cyclic Coverings Of P1 --- p.22 / Chapter 5.2 --- Abelian Varieties And CM-type --- p.24 / Chapter 6 --- Families of Jacobians Coming From [6] --- p.27 / Chapter 6.1 --- Example 1. Family y3 = x(x ´ؤ l)(x ´ؤ λ)(x ´ؤμ) --- p.27 / Chapter 6.2 --- Example 2. Family y5=x(x ´ؤ l)(x ´ؤ λ)(x ´ؤμ) --- p.34 / Chapter 6.3 --- Other Families --- p.38 / Bibliography --- p.60

Huang Yong Dong. / Thesis submitted in: November 2003. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 59-60). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- Abelian Varieties And Shimura Varieties --- p.5 / Chapter 1.2 --- Jacobians of Some Families of Curves --- p.7 / Chapter 1.3 --- Endomorphism Algebras of Jacobians of Curves --- p.8 / Chapter 2 --- Families of Abelian Varieties --- p.11 / Chapter 2.1 --- Abelian Varieties --- p.11 / Chapter 2.2 --- The Endomorphism Algebra of A Simple Abelian Varieties --- p.13 / Chapter 2.3 --- Family of Abelian Varieties and Shimura Varieties --- p.15 / Chapter 2.3.1 --- Real Multiplication --- p.16 / Chapter 2.3.2 --- Totally Indefinite Quaternion Multiplication --- p.19 / Chapter 2.3.3 --- Totally Definite Quaternion Multiplication --- p.22 / Chapter 2.3.4 --- Complex Multiplication --- p.25 / Chapter 2.3.5 --- Shimura Varieties --- p.28 / Chapter 2.4 --- The Endomorphism Algebra of A General Member --- p.29 / Chapter 3 --- Jacobians of Some Families of Curves --- p.32 / Chapter 3.1 --- Some Families of Curves --- p.32 / Chapter 3.2 --- Kodaira-Spencer Map --- p.37 / Chapter 3.3 --- Infinity of CM Type Points --- p.46 / Chapter 3.3.1 --- Φ Is Dominant --- p.46 / Chapter 3.3.2 --- Infinity of CM Points --- p.50 / Chapter 4 --- Endomorphism Algebras of Jacobians of Some Fam- ilies of Curves --- p.51 / Chapter 4.1 --- Jacobians Between Finite Coverings of Curves --- p.51 / Chapter 4.2 --- Endomorphism Algebras of Families of Jacobians --- p.53 / Chapter 4.2.1 --- The Case For μ = 1 --- p.53 / Chapter 4.2.2 --- The Case For μ = 2 --- p.55 / Chapter 4.2.3 --- The Case For μ =3 --- p.58 / Bibliography

The projective plane, PG(2;q), over a Galois field Fq is an incidence structure of points and lines. A (k;n)-arc K in PG(2;q) is a set of k points such that no n+1 of them are collinear but some n are collinear. A (k;n)-arc K in PG(2;q) is called complete if it is not contained in any (k+1;n)-arc. The existence of arcs for particular values of k and n pose interesting problems in finite geometry. It connects with coding theory and graph theory, with important applications in computer science. The main problem, known as the packing problem, is to determine the largest size mn(2;q) of K in PG(2;q). This problem has received much attention. Here, the work establishes complete arcs with a large number of points. In contrast, the problem to determine the smallest size tn(2;q) of a complete (k;n)-arc is mostly based on the lower bound arising from theoretical investigations. This thesis has several goals. The first goal is to classify certain (k;4)-arcs for k = 6,...,38 in PG(2;13). This classification is established through an approach in Chapter 2. This approach uses a new geometrical method; it is a combination of projective inequivalence of (k;4)-arcs up to k = 6 and certain sdinequivalent (k;4)-arcs that have sd-inequivalent classes of secant distributions for k = 7,...,38. The part related to projectively inequivalent (k;4)-arcs up to k=6 starts by fixing the frame points f1;2;3;88g and then classify the projectively inequivalent (5;4)-arcs. Among these (5;4)-arcs and (6;4)-arcs, the lexicographically least set are found. Now, the part regarding sd-inequivalent (k;4)-arcs in this method starts by choosing five sd-inequivalent (7;4)-arcs. This classification method may not produce all sd-inequivalent classes of (k;4)-arcs. However, it was necessary to employ this method due to the increasing number of (k;4)-arcs in PG(2;13) and the extreme computational difficulty of the problem. It reduces the constructed number of (k;4)-arcs in each process for large k. Consequently, it reduces the executed time for the computation which could last for years. Also, this method decreases the memory usage needed for the classification. The largest size of (k;4)-arc established through this method is k = 38. The classification of certain (k;4)-arcs up to projective equivalence, for k = 34,35,36,37,38, is also established. This classification starts from the 77 incomplete (34;4)-arcs that are constructed from the sd-inequivalent (33;4)-arcs given in Section 2.29, Table 2.35. Here, the largest size of (k;4)-arc is still k = 38. In addition, the previous process is re-iterated with a different choice of five sd-inequivalent (7;4)-arcs. The purpose of this choice is to find a new size of complete (k;4)-arc for k > 38. This particular computation of (k;4)-arcs found no complete (k;4)-arc for k > 38. In contrast, a new size of complete (k;4)-arc in PG(2;13) is discovered. This size is k = 36 which is the largest complete (k;4)-arc in this computation. This result raises the second largest size of complete (k;4)-arc found in the first classification from k = 35 to k = 36. The second goal is to discuss the incidence structure of the orbits of the groups of the projectively inequivalent (6;4)-arcs and also the incidence structures of the orbits of the groups other than the identity group of the sd-inequivalent (k;4)-arcs. In Chapter 3, these incidence structures are given for k = 6,7,8,9,10,11,12,13,14,38. Also, the pictures of the geometric configurations of the lines and the points of the orbits are described. The third goal is to find the sizes of certain sd-inequivalent complete (k;4)-arcs in PG(2;13). These sizes of complete (k;4)-arcs are given in Chapter 4 where the smallest size of complete (k;4)-arc is at most k = 24 and the largest size is at least k = 38. The fourth goal is to give an example of an associated non-singular quartic curve C for each complete (k;4)-arc and to discuss the algebraic properties of each curve in terms of the number I of inflexion points, the number jC \K j of rational points on the corresponding arc, and the number N1 of rational points of C . These curves are given in Chapter 5. Also, the algebraic properties of complete arcs of the most interesting sizes investigated in this thesis are studied. In addition, there are two examples of quartic curves C (g0 1) and C (g0 2) attaining the Hasse-Weil- Serre upper bound for the number N1 of rational points on a curve over the finite field of order thirteen. This number is 32. The fifth goal is to classify the (k;4)-arcs in PG(2;13) up to projective inequivalence for k < 10. This classification is established in Chapter 6. It starts by fixing a triad, U1, on the projective line, PG(1;13). Here, the number of projectively inequivalent (k;4)-arcs are tested by using the tool given in Chapter 2. Then, among the number of the projectively inequivalent (10;4)-arcs found, the classification of sd-inequivalent (k;4)-arcs for k = 10 is made. The number of these sd-inequivalent arcs is 36. Then, the 36 sd-inequivalent arcs are extended. The aim here is to investigate if there is a new size of sd-inequivalent (k;4)-arc for k > 38 that can be obtained from these arcs. The largest size of sd-inequivalent (k;4)-arc in this process is the same as the largest size of the sd-inequivalent (k;4)-arc established in Chapter 2, that is, k = 38. In addition, the classification of (k;n)-arcs in PG(2;13) is extended from n = 4 to n = 6. This extension is given in Chapter 7 where some results of the classification of certain (k;6)-arcs for k = 9; : : : ;25 are obtained using the same method as in Chapter 2 for k = 7,...,38. This process starts by fixing a certain (8;6)-arc containing six collinear points in PG(2;13).