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E-raamat: Designs From Linear Codes

(Hong Kong Univ Of Sci & Tech, Hong Kong)
  • Formaat: 392 pages
  • Ilmumisaeg: 24-Aug-2018
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813274341
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Designs From Linear CodesSuurem pilt
  • Formaat: 392 pages
  • Ilmumisaeg: 24-Aug-2018
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813274341
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This monograph aims to provide a well-rounded and detailed account of designs using linear codes. Most chapters of this monograph cover on the designs of linear codes. A few chapters deal with designs obtained from linear codes in other ways. Connections among ovals, hyperovals, maximal arcs, ovoids, linear codes and designs are also investigated. This book consists of both classical results on designs from linear codes and recent results yet published by others. This monograph is intended to be a reference for postgraduates and researchers who work on combinatorics, or coding theory, or digital communications, or finite geometry.

Preface v
1 Mathematical Foundations
1(62)
1.1 The Rings Zn
1(1)
1.2 Finite Fields
2(8)
1.2.1 Introduction to Finite Fields
2(1)
1.2.2 Traces, Norms, and Bases
3(1)
1.2.3 Field Automorphisms
4(1)
1.2.4 Additive and Multiplicative Characters
4(2)
1.2.5 Several Types of Character Sums
6(3)
1.2.6 Quadratic Forms over GF{q)
9(1)
1.3 Group Algebra
10(1)
1.4 Special Types of Polynomials
10(6)
1.4.1 Permutation Polynomials over Finite Fields
10(1)
1.4.2 Dickson Polynomials over Finite Fields
11(1)
1.4.3 Krawtchouk Polynomials
12(4)
1.5 Cyclotomy in GF(r)
16(4)
1.5.1 Cyclotomy
16(2)
1.5.2 Cyclotomy in GF(r)
18(2)
1.6 Basics of Group Actions
20(4)
1.7 Permutation Groups and Their Actions
24(14)
1.7.1 Semilinear Mappings of GF{q)m
24(1)
1.7.2 General Linear Groups GLm(GF(q))
25(1)
1.7.3 General Semilinear Groups ΓLm(GF(q))
26(1)
1.7.4 Special Linear Groups SLm(GF(q))
27(1)
1.7.5 General Affine Groups GAm(GF(q))
28(1)
1.7.6 Special Affine Groups SAm(GF(q))
29(1)
1.7.7 Semilinear Affine Groups ΓAm(GF(q))
29(1)
1.7.8 Projective General Linear Groups PGLm(GF(q))
30(3)
1.7.9 Projective Semilinear Groups PΓLm(GF(q))
33(1)
1.7.10 Projective Special Linear Groups PSLm(GF(q))
33(2)
1.7.11 A Summary of the Group Actions on GF(q)m and (GF(q)m)*
35(1)
1.7.12 Permutation Group Actions on GF(qm) and GF(qm)*
36(1)
1.7.13 Highly Transitive Permutation Groups
36(1)
1.7.14 Homogeneous Permutation Groups
37(1)
1.8 Finite Geometries
38(11)
1.8.1 Projective Spaces PG(m, GF(q))
38(1)
1.8.2 Affine Spaces AG(m, GF(q))
39(1)
1.8.3 Projective Planes
40(2)
1.8.4 Desarguesian Projective Planes PG(2, GF(q))
42(2)
1.8.5 Central Collineations and Homologies of Projective Planes
44(2)
1.8.6 Affine Planes
46(3)
1.9 Planar Functions
49(5)
1.9.1 Definitions and Properties
49(1)
1.9.2 Some Known Planar Functions
50(1)
1.9.3 Planar Functions from Semifields
51(3)
1.9.4 Affine Planes from Planar Functions
54(1)
1.10 Almost Perfect Nonlinear and Almost Bent Functions
54(2)
1.10.1 APN Functions
54(1)
1.10.2 AB Functions
55(1)
1.11 Periodic Sequences
56(2)
1.11.1 The Linear Span
56(1)
1.11.2 Correlation Functions
57(1)
1.12 Difference Sets
58(5)
1.12.1 Fundamentals of Difference Sets
58(2)
1.12.2 Divisible and Relative Difference Sets
60(1)
1.12.3 Characteristic Sequence of Difference Sets in Zn
61(1)
1.12.4 Characteristic Functions of Difference Sets
61(2)
2 Linear Codes over Finite Fields
63(26)
2.1 Linear Codes over GF(q)
63(2)
2.2 The Mac Williams Identity and Transform
65(2)
2.3 The Pless Power Moments
67(1)
2.4 Punctured Codes of a Linear Code
68(1)
2.5 Shortened Codes of a Linear Code
69(1)
2.6 Extended Code of a Linear Code
69(2)
2.7 Augmented Code of a Linear Code
71(1)
2.8 Automorphism Groups and Equivalences of Linear Codes
71(3)
2.9 Subfield Subcodes
74(2)
2.10 Bounds on the Size of Linear Codes
76(4)
2.11 Restrictions on Parameters of Linear Codes
80(1)
2.12 Bounds on the Size of Constant Weight Codes
80(1)
2.13 Hamming and Simplex Codes
81(1)
2.14 A Trace Construction of Linear Codes
82(2)
2.15 Projective Linear Codes and Projective Geometry
84(5)
3 Cyclic Codes over Finite Fields
89(22)
3.1 Factorization of xn --- 1 over GF(q)
89(1)
3.2 Generator and Parity Check Polynomials
90(2)
3.3 Idempotents of Cyclic Codes
92(2)
3.4 Zeros of Cyclic Codes
94(2)
3.5 Lower Bounds on the Minimum Distance
96(1)
3.6 BCH Codes
97(3)
3.6.1 Definition and Basic Properties
97(3)
3.6.2 Recent Advances in BCH Codes
100(1)
3.7 Quadratic Residue Codes
100(7)
3.7.1 Quadratic Residue Codes
100(3)
3.7.2 Extended Quadratic Residue Codes
103(4)
3.8 Duadic Codes
107(1)
3.9 Irreducible Cyclic Codes
108(1)
3.10 A Combinatorial Approach to Cyclic Codes
108(3)
4 Designs and Codes
111(20)
4.1 Fundamentals of t-Designs
111(6)
4.1.1 Incidence Structures
111(1)
4.1.2 Incidence Matrices
112(1)
4.1.3 Isomorphisms and Automorphisms
112(1)
4.1.4 Definition and Properties of t-Designs
113(3)
4.1.5 Intersection Numbers of Designs
116(1)
4.1.6 Complementary, Derived and Residual Designs
116(1)
4.2 The Classical Codes of Designs
117(3)
4.2.1 Linear Codes of Incidence Structures
117(1)
4.2.2 The Classical Codes of Designs
118(2)
4.3 The Support Designs of Linear Codes
120(6)
4.3.1 The Construction of t-Designs from Linear Codes
120(5)
4.3.2 MDS Codes and Complete Designs
125(1)
4.3.3 Constructing Designs from Related Binary Codes
125(1)
4.4 Designs of Codes with Special Automorphism Groups
126(1)
4.5 Designs from Finite Geometries
127(4)
5 Designs of Binary Reed-Muller Codes
131(24)
5.1 Binary Reed-Muller Codes and Their Relatives
131(10)
5.2 Designs from the Binary Reed-Muller Codes
141(10)
5.2.1 Designs in R2(1,m) and R2(m --- 2,m)
142(2)
5.2.2 Designs in R2(2,m) and R2(m --- 3,m)
144(6)
5.2.3 Designs in R2(r,m) for 3 ≤ r ≤ m --- 4
150(1)
5.2.4 Designs from Binary Codes between R2(r,m) and R2(r+1,m)
151(1)
5.3 Designs from the Punctured Binary Reed-Muller Codes
151(4)
6 Affine Invariant Codes and Their Designs
155(34)
6.1 Affine Invariant Codes
155(6)
6.2 Designs of Affine-Invariant Codes
161(1)
6.3 Specific Families of Affine-Invariant Codes and Their Designs
162(26)
6.3.1 Extended Narrow-Sense Primitive BCH Codes
162(1)
6.3.2 Generalised Reed-Muller Codes and Their Designs
163(7)
6.3.3 Dilix Codes and Their Designs
170(9)
6.3.4 Extended Binary Cyclic Codes with Zeros of the Forms α and α1+2e and Their Designs
179(9)
6.4 Notes on Affine-Invariant Codes
188(1)
7 Weights in Some BCH Codes over GF(q)
189(22)
7.1 A Recall of BCH Codes
189(1)
7.2 The Parameters of the Codes C(q,qm--1,δ1, 1) and C(q,qm--1,δ1,0), where δ1 = (q -- 1)qm--1 --- 1
189(1)
7.3 The Parameters of the Codes C(q,qm--1,δ2, 1) and C(q,qm--1,δ2,0), where δ2 = (q -- 1)qm--1 --- 1--- q(m--1)/2
190(12)
7.4 The Parameters of the Codes C(q,qm--1,δ3, 1) and C(q,qm--1,δ3,0), where δ3 = (q --- 1)am--1 --- 1 --- q(m--1)/1
202(8)
7.5 Weights in C(2,2m--1,δ,1) and Its Dual for δ ∈ {3,5,7}
210(1)
8 Designs from Four Types of Linear Codes
211(30)
8.1 Designs from a Type of Binary Codes with Three Weights
211(9)
8.2 Designs from a Type of Binary Codes with Five Weights
220(14)
8.2.1 The Codes with Five Weights and Their Related Codes
220(10)
8.2.2 Infinite Families of 2-Designs from Cm and Cm
230(2)
8.2.3 Infinite Families of 3-Designs from Cm and Cm
232(2)
8.2.4 Two Families of Binary Cyclic Codes with the Weight Distribution of Table 8.2
234(1)
8.3 Infinite Families of Designs from a Type of Ternary Codes
234(4)
8.4 Conjectured Infinite Families of Designs from Cyclic Codes
238(3)
9 Designs from Primitive BCH Codes
241(16)
9.1 A General Theorem on Designs from Primitive BCH Codes
241(1)
9.2 Designs from the Primitive BCH Codes C(2,2m--1,δ2,1)
242(3)
9.3 Designs from the Primitive BCH Codes C(q,qm--1,δ2,1) for odd prime q
245(2)
9.4 Designs from the Primitive BCH Codes C(2,2m--1,δ3,1))
247(3)
9.5 Designs from the Primitive BCH Codes C(q,qm--1,δ3,1) for q
250(1)
9.6 Designs from C(2,2m--1,5,1) C(2,2m--1,5,1)⊥ for even m ≥ 4
251(3)
9.7 Designs from the Primitive BCH Codes C(q,qm--1,3,1) for q ≥ 3
254(3)
10 Designs from Codes with Regularity
257(24)
10.1 Packing and Covering Radii
257(1)
10.2 The Characteristic Polynomial of a Code
258(5)
10.3 Regular Codes and Their Designs
263(3)
10.4 Perfect Codes
266(2)
10.5 Designs in Perfect Codes
268(7)
10.5.1 Theory of Designs in Perfect Codes
268(1)
10.5.2 Designs in the [ 23, 12, 7] Golay Binary Code
269(1)
10.5.3 Designs in the [ 11, 6, 5] Golay Ternary Code
270(1)
10.5.4 Designs in the Hamming and Simplex Codes
271(4)
10.6 Designs in Uniformly Packed Codes
275(6)
10.6.1 Definitions, Properties and General Results
275(2)
10.6.2 Designs in Uniformly Packed Binary Codes
277(4)
11 Designs from QR and Self-Dual Codes
281(14)
11.1 Self-Dual Codes and Their Designs
281(6)
11.1.1 Definition and Existence
281(2)
11.1.2 Weight Enumerators of Self-Dual Codes
283(2)
11.1.3 Extremal Self-Dual Codes and Their Designs
285(2)
11.2 Designs from Extended Quadratic Residue Codes
287(3)
11.2.1 Infinite Families of 2-Designs and 3-Designs
287(1)
11.2.2 Sporadic 5-Designs from Self-Dual Codes
287(3)
11.3 Pless Symmetry Codes and Their Designs
290(4)
11.4 Other Self-Dual Codes Holding t-Designs
294(1)
12 Designs from Arc and MDS Codes
295(24)
12.1 Arcs, Caps, Conies, Hyperovals and Ovals in PG(2, GF(g))
295(4)
12.2 Hyperovals in PG(2, GF(q)) and [ q + 2, 3, q] MDS Codes
299(1)
12.3 Oval Polynomials on GF(2m)
300(4)
12.3.1 Basic Properties of Oval Polynomials
300(1)
12.3.2 Translation Oval Polynomials
301(1)
12.3.3 Segre and Glynn Oval Polynomials
301(1)
12.3.4 Cherowitzo Oval Polynomials
302(1)
12.3.5 Payne Oval Polynomials
303(1)
12.3.6 Subiaco Oval Polynomials
303(1)
12.4 A Family of Hyperovals from Extended Cyclic Codes
304(1)
12.5 Hyperoval Designs
305(2)
12.6 Hadamard Designs from Hyperovals
307(3)
12.7 Maximal Arc Codes and Their Designs
310(3)
12.8 A Family of Extended Cyclic Codes and Their Designs
313(6)
13 Designs from Oviod Codes
319(12)
13.1 Ovoids in PG(3,GF(g)) and Their Properties
319(1)
13.2 Ovoids in PG(3, GF(q)) and [ q2 + 1, 4, q2 --- q] Codes
320(1)
13.3 A Family of Cyclic Codes with Parameters [ q2 + 1, 4, q2 --- q]
321(3)
13.4 Designs from Ovoid Codes over GF(q)
324(4)
13.5 Ovoids, Codes, Designs and Inversive Planes
328(3)
14 Quasi-Symmetric Designs from Bent Codes
331(24)
14.1 Derived and Residual Designs of Symmetric Designs
331(1)
14.2 Symmetric and Quasi-symmetric SDP Designs
332(1)
14.3 The Roadmap of the Remaining Sections
332(2)
14.4 Bent Functions
334(1)
14.5 Symmetric 2-(2m, 2m--1 --- 2m--2/2, 2m--2 ---2m--2/2) Designs and Their Codes
335(1)
14.6 Symmetric 2-(2m, 2m--1 --- 2m--2/2, 2m--2 --- 2m--2/2) SDP Designs
336(5)
14.7 Derived and Residual Designs of Symmetric SDP Designs
341(3)
14.8 A General Construction of Linear Codes with Bent Functions
344(4)
14.9 Infinite Families of 2-Designs from Bent Codes
348(4)
14.10 Concluding Notes
352(3)
Appendix A Designs from Binary Codes with Regularities 355(6)
A.1 Four Fundamental Parameters of Codes
355(2)
A.2 Designs from Codes with Regularity
357(4)
A.2.1 Designs from Codes When s ≤ d1
358(1)
A.2.2 Designs from Nonlinear Codes When s1 ≤ d
359(2)
Bibliography 361(12)
Index 373
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