Self-dual codes and invariant theory: with 34 tables
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2006
|
| Series: | Algorithms and computation in mathematics
17 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Auch als Internetausgabe |
| Physical Description: | XXVI, 430 S. graph. Darst. |
| ISBN: | 9783540307297 354030729X |
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MARC
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| 100 | 1 | |a Nebe, Gabriele |d 1967- |e Verfasser |0 (DE-588)17285671X |4 aut | |
| 245 | 1 | 0 | |a Self-dual codes and invariant theory |b with 34 tables |c Gabriele Nebe ; Eric M. Rains ; Neil J. A. Sloane |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XXVI, 430 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Algorithms and computation in mathematics |v 17 | |
| 500 | |a Auch als Internetausgabe | ||
| 650 | 4 | |a Coding theory | |
| 650 | 7 | |a Configurações combinatórias. |2 larpcal | |
| 650 | 4 | |a Duality theory (Mathematics) | |
| 650 | 4 | |a Invariants | |
| 650 | 7 | |a Teoria dos códigos. |2 larpcal | |
| 650 | 0 | 7 | |a Selbstdualität |0 (DE-588)4180824-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Invariantentheorie |0 (DE-588)4162209-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Code |0 (DE-588)4010345-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Code |0 (DE-588)4010345-6 |D s |
| 689 | 0 | 1 | |a Selbstdualität |0 (DE-588)4180824-1 |D s |
| 689 | 0 | 2 | |a Invariantentheorie |0 (DE-588)4162209-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Rains, Eric M. |e Verfasser |4 aut | |
| 700 | 1 | |a Sloane, Neil J. A. |d 1939- |e Verfasser |0 (DE-588)121291553 |4 aut | |
| 830 | 0 | |a Algorithms and computation in mathematics |v 17 |w (DE-604)BV011131286 |9 17 | |
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Record in the Search Index
| _version_ | 1804135213893156864 |
|---|---|
| adam_text | GABRIELE NEBE
ERIC M. RAINS
NEIL J.A. SLOANE
SELF-DUAL CODES
AND INVARIANT THEORY
WITH 10 FIGURES AND 34 TABLES
4Y SPRINGER
CONTENTS
PREFACE V
LIST OF SYMBOLS XIV
LIST OF TABLES XXV
LIST OF FIGURES XXVII
1 THE TYPE OF A SELF-DUAL CODE 1
1.1 QUADRATIC MAPS 2
1.2 SELF-DUAL AND ISOTROPIC CODES 4
1.3 TWISTED MODULES AND THEIR REPRESENTATIONS 5
1.4 TWISTED RINGS AND THEIR REPRESENTATIONS 6
1.5 TRIANGULAR TWISTED RINGS 9
1.6 QUADRATIC PAIRS AND THEIR REPRESENTATIONS 11
1.7 FORM RINGS AND THEIR REPRESENTATIONS 13
1.8 THE TYPE OF A CODE 15
1.9 TRIANGULAR FORM RINGS 18
1.10 MATRIX RINGS OF FORM RINGS AND THEIR REPRESENTATIONS 19
1.11 AUTOMORPHISM GROUPS OF CODES 22
1.12 SHADOWS 24
2 WEIGHT ENUMERATORS AND IMPORTANT TYPES 29
2.1 WEIGHT ENUMERATORS OF CODES 29
2.2 MAC WILLIAMS IDENTITY AND GENERALIZATIONS 35
2.2.1 THE WEIGHT ENUMERATOR OF THE SHADOW 39
2.3 CATALOGUE OF IMPORTANT TYPES 39
2.3.1 BINARY CODES 40
2 40
2! 41
2N 41
2S 41
2.3.2 EUCLIDEAN CODES 42
4
E
42
XVIII CONTENTS
Q
E
(EVEN) 43
$ 44
3 45
Q
E
(ODD) 46
QF (ODD) 46
2.3.3 HERMITIAN CODES 47
4
H
47
Q
11
47
FL 48
2.3.4 ADDITIVE CODES 48
4
H+
48
Q
H
+ (EVEN) 49
G?+ (EVEN) 49
Q%
+
(EVEN) 50
& (EVEN) 50
Q
B+
(ODD) 50
QF+ (ODD) 51
2.3.5 CODES OVER GALOIS RINGS Z/MZ 51
4
Z
52
M
Z
53
M 54
M
X
54
MFJ
X
55
M§ 55
2.3.6 CODES OVER MORE GENERAL GALOIS RINGS 55
GRQ/,/)
E
55
GR(P
P
,F)F 56
GR(P
E
,/)
E
56
GR(2
E
,/)| 57
GR(2
E
,/)P! 57
GR(2
E
,/)
E2
58
GR(P
E
,/)
H
58
GR(P
E
,/), 58
/)
H
+ 59
F)F+ 59
2.3.7 LINEAR CODES OVER P-ADIC INTEGERS 60
Z
P
60
MORE GENERAL P-ADIC INTEGERS 60
2.4 EXAMPLES OF SELF-DUAL CODES 60
2.4.1 2: BINARY CODES 60
2J: SINGLY-EVEN BINARY SELF-DUAL CODES 61
2JI: DOUBLY-EVEN BINARY SELF-DUAL CODES 61
2.4.2 4
E
: EUCLIDEAN SELF-DUAL CODES OVER F
4
64
2.4.3 Q
E
(EVEN OR ODD): EUCLIDEAN SELF-DUAL CODES OVER
Q
. . . . 65
CONTENTS XIX
2.4.4 (/{P GENERALIZED DOUBLY-EVEN SELF-DUAL CODES 65
2.4.5 3: EUCLIDEAN SELF-DUAL CODES OVER F3 67
2.4.6 4
H
: HERMITIAN SELF-DUAL CODES OVER F4 68
2.4.7 Q
H
: HERMITIAN SELF-DUAL LINEAR CODES OVER
Q
68
2.4.8 4
H+
: TRACE-HERMITIAN ADDITIVE CODES OVER F4 69
2.4.9 4
Z
: SELF-DUAL CODES OVER Z/4Z 70
2.4.10 CODES OVER OTHER GALOIS RINGS 76
2.4.11 Z
P
: CODES OVER THE P-ADIC NUMBERS 77
2.5 THE GLEASON-PIERCE THEOREM 80
CLOSED CODES 83
3.1 BILINEAR FORMS AND CLOSED CODES 83
3.2 FAMILIES OF CLOSED CODES 86
3.2.1 CODES OVER COMMUTATIVE RINGS 88
3.2.2 CODES OVER QUASI-FROBENIUS RINGS 89
3.2.3 ALGEBRAS OVER A COMMUTATIVE RING 90
3.2.4 DIRECT SUMMANDS 94
3.3 REPRESENTATIONS OF TWISTED RINGS AND CLOSED CODES 94
3.4 MORITA THEORY 96
3.5 NEW REPRESENTATIONS FROM OLD 98
3.5.1 SUBQUOTIENTS AND QUOTIENTS 98
3.5.2 DIRECT SUMS AND PRODUCTS 99
3.5.3 TENSOR PRODUCTS 100
THE CATEGORY QUAD 103
4.1 THE CATEGORY OF QUADRATIC GROUPS 104
4.2 THE INTERNAL HOM-FUNCTOR IHOM 108
4.3 PROPERTIES OF QUADRATIC RINGS 113
4.4 MORITA THEORY FOR QUADRATIC RINGS 116
4.5 MORITA THEORY FOR FORM RINGS 120
4.6 WITT RINGS, GROUPS AND MODULES 121
THE MAIN THEOREMS 129
5.1 PARABOLIC GROUPS 130
5.2 HYPERBOLIC CO-UNITARY GROUPS 131
5.2.1 GENERATORS FOR THE HYPERBOLIC CO-UNITARY GROUP 136
5.3 CLIFFORD-WEIL GROUPS 139
5.4 SCALAR ELEMENTS IN C(P) 142
5.5 CLIFFORD-WEIL GROUPS AND FULL WEIGHT ENUMERATORS 149
5.6 RESULTS FROM INVARIANT THEORY 155
5.6.1 MOLIEN SERIES 155
5.6.2 RELATIVE INVARIANTS 158
5.6.3 CONSTRUCTION OF INVARIANTS USING DIFFERENTIAL OPERATORS . . 160
5.6.4 INVARIANTS AND DESIGNS 161
5.7 SYMMETRIZATIONS 162
CONTENTS
5.8 EXAMPLE: HERMITIAN CODES OVER FG 167
REAL AND COMPLEX CLIFFORD GROUPS 171
6.1 BACKGROUND 171
6.2 RUNGE S THEOREMS 174
6.3 THE REAL CLIFFORD GROUP C
M
177
6.4 THE COMPLEX CLIFFORD GROUP X
M
182
6.5 BARNES-WALL LATTICES 184
6.6 MAXIMAL FINITENESS IN REAL CASE 188
6.7 MAXIMAL FINITENESS IN COMPLEX CASE 190
6.8 AUTOMORPHISM GROUPS OF WEIGHT ENUMERATORS 190
CLASSICAL SELF-DUAL CODES 193
7.1 QUASISIMPLE FORM RINGS 193
7.2 SPLIT TYPE 195
7.2.1 Q
IIN
: LINEAR CODES OVER F, 196
CLIFFORD-WEIL GROUPS 198
F
2
, GENUS 1 198
F
2
, GENUS 2 199
7.3 HERMITIAN TYPE 201
7.3.1 Q
H
: HERMITIAN SELF-DUAL CODES OVER F
G
202
CLIFFORD-WEIL GROUPS 202
THE CASE Q = 4 203
THE CASE Q = 9 206
7.4 ORTHOGONAL (OR EUCLIDEAN) TYPE, P ODD 207
7.4.1 Q
E
(ODD): EUCLIDEAN SELF-DUAL CODES OVER
Q
207
CLIFFORD-WEIL GROUPS (Q ODD) 207
THE CASE Q = 3 209
THE CASE Q = 3, GENUS 2 210
THE CASE Q = 9 211
THE CASE Q = 5 212
7.5 SYMPLECTIC TYPE, P ODD 213
7.5.1 G
H+
(ODD): HERMITIAN F
R
-LINEAR CODES OVER
Q
, Q = R
2
. . 214
CLIFFORD-WEIL GROUPS (GENUS G) 214
THE CASE Q = 9, GENUS 1 215
7.6 CHARACTERISTIC 2, ORTHOGONAL AND SYMPLECTIC TYPES 215
7.6.1 Q
U+
(EVEN): HERMITIAN F
R
-LINEAR CODES OVER
Q
, Q = R
2
. . 217
CLIFFORD-WEIL GROUPS (GENUS G) 217
THE CASE Q = 4, GENUS 1 217
THE CASE Q = A, GENUS 2 219
THE CASE Q = 16 220
7.6.2 Q
E
(EVEN): EUCLIDEAN SELF-DUAL F
9
-LINEAR CODES 220
CLIFFORD-WEIL GROUPS (GENUS G) 220
THE CASE Q = 2 221
THE CASE Q = A 221
CONTENTS XXI
7.6.3 Q^
+
(EVEN): EVEN TRACE-HERMITIAN F
R
-LINEAR CODES 222
CLIFFORD-WEIL GROUPS (GENUS G) 222
THE CASE Q = 4, GENUS 1 223
7.6.4 Q^ (EVEN): GENERALIZED DOUBLY-EVEN CODES OVER
Q
224
CLIFFORD-WEIL GROUPS (GENUS G) 224
THE CASE K = 2, ARBITRARY GENUS 225
THE CASE K = F
4
, GENUS 1 225
THE CASE K = F
8
226
FURTHER EXAMPLES OF SELF-DUAL CODES 227
8.1 M
Z
: CODES OVER Z/RNZ 227
8.2 4
Z
: SELF-DUAL CODES OVER Z/4Z 230
8.2.1 4
Z
: TYPE I SELF-DUAL CODES OVER Z/4Z 230
8.2.2 4
Z
: TYPE I SELF-DUAL CODES OVER Z/4Z CONTAINING 1 231
8.2.3 SAME, WITH 1 IN THE SHADOW 233
8.2.4 4F
T
: TYPE II SELF-DUAL CODES OVER Z/4Z 233
8.2.5 4F
U
: TYPE II SELF-DUAL CODES OVER Z/4Z CONTAINING 1 ... 234
8.3 8
Z
: SELF-DUAL CODES OVER Z/8Z 234
8.4 CODES OVER MORE GENERAL GALOIS RINGS 235
8.4.1 GR(P
E
, /)
E
: EUCLIDEAN SELF-DUAL GR(P
E
, /)-LINEAR CODES. . 236
8.4.2 GR(P
E
, /)
H
: HERMITIAN SELF-DUAL GR(P
E
, /)-LINEAR CODES. 238
8.4.3 GR(P
E
, 2Z)
H+
: TRACE-HERMITIAN GR(P
E
, I)-LINEAR CODES.. . 239
8.4.4 CLIFFORD-WEIL GROUPS FOR GR(4,2) 239
8.5 SELF-DUAL CODES OVER F,
2
+
Q
2 U 243
LATTICES 249
9.1 LATTICES AND THETA SERIES 252
9.1.1 PRELIMINARY DEFINITIONS 252
9.1.2 MODULAR LATTICES AND ATKIN-LEHNER INVOLUTIONS 255
9.1.3 SHADOWS 260
9.1.4 JACOBI FORMS 261
9.1.5 SIEGEL THETA SERIES 261
JACOBI-SIEGEL THETA SERIES AND RIEMANN THETA FUNCTIONS 265
RIEMANN THETA FUNCTIONS WITH HARMONIC COEFFICIENTS . . . 268
9.1.6 HILBERT THETA SERIES 269
9.2 POSITIVE DEFINITE FORM R-ALGEBRAS 272
9.3 HALF-SPACES 274
9.4 FORM ORDERS AND LATTICES 276
9.5 EVEN AND ODD UNIMODULAR LATTICES 278
9.6 GLUING THEORY FOR CODES 280
9.7 GLUING THEORY FOR LATTICES 282
XXII CONTENTS
10 MAXIMAL ISOTROPIC CODES AND LATTICES 285
10.1 MAXIMAL ISOTROPIC CODES 286
10.2 MAXIMAL ISOTROPIC DOUBLY-EVEN BINARY CODES 290
10.3 MAXIMAL ISOTROPIC EVEN BINARY CODES 293
10.4 MAXIMAL ISOTROPIC TERNARY CODES 293
10.5 MAXIMAL ISOTROPIC ADDITIVE CODES OVER F4 298
10.6 MAXIMAL ISOTROPIC CODES OVER Z/4Z 298
10.7 MAXIMAL EVEN LATTICES 301
10.7.1 MAXIMAL EVEN LATTICES OF DETERMINANT 3
FC
304
10.7.2 MAXIMAL EVEN AND INTEGRAL LATTICES OF DETERMINANT 2
K
. . 306
11 EXTREMAL AND OPTIMAL CODES 313
11.1 UPPER BOUNDS 314
11.1.1 EXTREMAL WEIGHT ENUMERATORS AND THE LP BOUND 314
11.1.2 SELF-DUAL BINARY CODES, 2
N
AND 2
R
317
11.1.3 SOME OTHER TYPES 321
11.1.4 A NEW DEFINITION OF EXTREMALITY 324
11.1.5 ASYMPTOTIC UPPER BOUNDS 326
11.2 LOWER BOUNDS 328
11.3 TABLES OF EXTREMAL SELF-DUAL CODES 331
11.3.1 BINARY CODES 331
11.3.2 TYPE 3: TERNARY CODES 336
11.3.3 TYPES 4
E
AND 4F
T
: EUCLIDEAN SELF-DUAL CODES OVER F
4
338
11.3.4 TYPE 4
H
: HERMITIAN LINEAR SELF-DUAL CODES OVER F4 339
11.3.5 TYPES 4
H+
AND 4
+
: TRACE-HERMITIAN CODES OVER F
4
. . . 340
11.3.6 TYPE 4
Z
: SELF-DUAL CODES OVER Z/4Z 342
11.3.7 OTHER TYPES 345
12 ENUMERATION OF SELF-DUAL CODES 347
12.1 THE MASS FORMULAE 347
12.2 ENUMERATION OF BINARY SELF-DUAL CODES 350
INTERRELATIONS BETWEEN TYPES 2I AND 2N 356
12.3 TYPE 3: TERNARY SELF-DUAL CODES 360
12.3.1 TYPES 4
E
AND 4F
X
: EUCLIDEAN SELF-DUAL CODES OVER F
4
363
12.4 TYPE 4
H
: HERMITIAN SELF-DUAL CODES OVER F4 363
12.5 TYPE 4
H+
: TRACE-HERMITIAN ADDITIVE CODES OVER F
4
365
12.6 TYPE 4
Z
: SELF-DUAL CODES OVER Z/4Z 366
12.7 OTHER ENUMERATIONS 367
13 QUANTUM CODES 369
13.1 DEFINITIONS 370
13.2 ADDITIVE AND SYMPLECTIC QUANTUM CODES 373
13.3 HAMMING WEIGHT ENUMERATORS 376
CONTENTS XXIII
13.4 LINEAR PROGRAMMING BOUNDS 381
13.5 OTHER ALPHABETS 382
13.6 A TABLE OF QUANTUM CODES 385
REFERENCES 391
INDEX 417
|
| adam_txt |
GABRIELE NEBE
ERIC M. RAINS
NEIL J.A. SLOANE
SELF-DUAL CODES
AND INVARIANT THEORY
WITH 10 FIGURES AND 34 TABLES
4Y SPRINGER
CONTENTS
PREFACE V
LIST OF SYMBOLS XIV
LIST OF TABLES XXV
LIST OF FIGURES XXVII
1 THE TYPE OF A SELF-DUAL CODE 1
1.1 QUADRATIC MAPS 2
1.2 SELF-DUAL AND ISOTROPIC CODES 4
1.3 TWISTED MODULES AND THEIR REPRESENTATIONS 5
1.4 TWISTED RINGS AND THEIR REPRESENTATIONS 6
1.5 TRIANGULAR TWISTED RINGS 9
1.6 QUADRATIC PAIRS AND THEIR REPRESENTATIONS 11
1.7 FORM RINGS AND THEIR REPRESENTATIONS 13
1.8 THE TYPE OF A CODE 15
1.9 TRIANGULAR FORM RINGS 18
1.10 MATRIX RINGS OF FORM RINGS AND THEIR REPRESENTATIONS 19
1.11 AUTOMORPHISM GROUPS OF CODES 22
1.12 SHADOWS 24
2 WEIGHT ENUMERATORS AND IMPORTANT TYPES 29
2.1 WEIGHT ENUMERATORS OF CODES 29
2.2 MAC WILLIAMS IDENTITY AND GENERALIZATIONS 35
2.2.1 THE WEIGHT ENUMERATOR OF THE SHADOW 39
2.3 CATALOGUE OF IMPORTANT TYPES 39
2.3.1 BINARY CODES 40
2 40
2! 41
2N 41
2S 41
2.3.2 EUCLIDEAN CODES 42
4
E
42
XVIII CONTENTS
Q
E
(EVEN) 43
$ 44
3 45
Q
E
(ODD) 46
QF (ODD) 46
2.3.3 HERMITIAN CODES 47
4
H
47
Q
11
47
FL 48
2.3.4 ADDITIVE CODES 48
4
H+
48
Q
H
+ (EVEN) 49
G?+ (EVEN) 49
Q%
+
(EVEN) 50
& (EVEN) 50
Q
B+
(ODD) 50
QF+ (ODD) 51
2.3.5 CODES OVER GALOIS RINGS Z/MZ 51
4
Z
52
M
Z
53
M\ 54
M\
X
54
MFJ
X
55
M§ 55
2.3.6 CODES OVER MORE GENERAL GALOIS RINGS 55
GRQ/,/)
E
55
GR(P
P
,F)F 56
GR(P
E
,/)
E
56
GR(2
E
,/)| 57
GR(2
E
,/)P! 57
GR(2
E
,/)
E2
58
GR(P
E
,/)
H
58
GR(P
E
,/), 58
/)
H
+ 59
F)F+ 59
2.3.7 LINEAR CODES OVER P-ADIC INTEGERS 60
Z
P
60
MORE GENERAL P-ADIC INTEGERS 60
2.4 EXAMPLES OF SELF-DUAL CODES 60
2.4.1 2: BINARY CODES 60
2J: SINGLY-EVEN BINARY SELF-DUAL CODES 61
2JI: DOUBLY-EVEN BINARY SELF-DUAL CODES 61
2.4.2 4
E
: EUCLIDEAN SELF-DUAL CODES OVER F
4
64
2.4.3 Q
E
(EVEN OR ODD): EUCLIDEAN SELF-DUAL CODES OVER
Q
. . . . 65
CONTENTS XIX
2.4.4 (/{P GENERALIZED DOUBLY-EVEN SELF-DUAL CODES 65
2.4.5 3: EUCLIDEAN SELF-DUAL CODES OVER F3 67
2.4.6 4
H
: HERMITIAN SELF-DUAL CODES OVER F4 68
2.4.7 Q
H
: HERMITIAN SELF-DUAL LINEAR CODES OVER
Q
68
2.4.8 4
H+
: TRACE-HERMITIAN ADDITIVE CODES OVER F4 69
2.4.9 4
Z
: SELF-DUAL CODES OVER Z/4Z 70
2.4.10 CODES OVER OTHER GALOIS RINGS 76
2.4.11 Z
P
: CODES OVER THE P-ADIC NUMBERS 77
2.5 THE GLEASON-PIERCE THEOREM 80
CLOSED CODES 83
3.1 BILINEAR FORMS AND CLOSED CODES 83
3.2 FAMILIES OF CLOSED CODES 86
3.2.1 CODES OVER COMMUTATIVE RINGS 88
3.2.2 CODES OVER QUASI-FROBENIUS RINGS 89
3.2.3 ALGEBRAS OVER A COMMUTATIVE RING 90
3.2.4 DIRECT SUMMANDS 94
3.3 REPRESENTATIONS OF TWISTED RINGS AND CLOSED CODES 94
3.4 MORITA THEORY 96
3.5 NEW REPRESENTATIONS FROM OLD 98
3.5.1 SUBQUOTIENTS AND QUOTIENTS 98
3.5.2 DIRECT SUMS AND PRODUCTS 99
3.5.3 TENSOR PRODUCTS 100
THE CATEGORY QUAD 103
4.1 THE CATEGORY OF QUADRATIC GROUPS 104
4.2 THE INTERNAL HOM-FUNCTOR IHOM 108
4.3 PROPERTIES OF QUADRATIC RINGS 113
4.4 MORITA THEORY FOR QUADRATIC RINGS 116
4.5 MORITA THEORY FOR FORM RINGS 120
4.6 WITT RINGS, GROUPS AND MODULES 121
THE MAIN THEOREMS 129
5.1 PARABOLIC GROUPS 130
5.2 HYPERBOLIC CO-UNITARY GROUPS 131
5.2.1 GENERATORS FOR THE HYPERBOLIC CO-UNITARY GROUP 136
5.3 CLIFFORD-WEIL GROUPS 139
5.4 SCALAR ELEMENTS IN C(P) 142
5.5 CLIFFORD-WEIL GROUPS AND FULL WEIGHT ENUMERATORS 149
5.6 RESULTS FROM INVARIANT THEORY 155
5.6.1 MOLIEN SERIES 155
5.6.2 RELATIVE INVARIANTS 158
5.6.3 CONSTRUCTION OF INVARIANTS USING DIFFERENTIAL OPERATORS . . 160
5.6.4 INVARIANTS AND DESIGNS 161
5.7 SYMMETRIZATIONS 162
CONTENTS
5.8 EXAMPLE: HERMITIAN CODES OVER FG 167
REAL AND COMPLEX CLIFFORD GROUPS 171
6.1 BACKGROUND 171
6.2 RUNGE'S THEOREMS 174
6.3 THE REAL CLIFFORD GROUP C
M
177
6.4 THE COMPLEX CLIFFORD GROUP X
M
182
6.5 BARNES-WALL LATTICES 184
6.6 MAXIMAL FINITENESS IN REAL CASE 188
6.7 MAXIMAL FINITENESS IN COMPLEX CASE 190
6.8 AUTOMORPHISM GROUPS OF WEIGHT ENUMERATORS 190
CLASSICAL SELF-DUAL CODES 193
7.1 QUASISIMPLE FORM RINGS 193
7.2 SPLIT TYPE 195
7.2.1 Q
IIN
: LINEAR CODES OVER F, 196
CLIFFORD-WEIL GROUPS 198
F
2
, GENUS 1 198
F
2
, GENUS 2 199
7.3 HERMITIAN TYPE 201
7.3.1 Q
H
: HERMITIAN SELF-DUAL CODES OVER F
G
202
CLIFFORD-WEIL GROUPS 202
THE CASE Q = 4 203
THE CASE Q = 9 206
7.4 ORTHOGONAL (OR EUCLIDEAN) TYPE, P ODD 207
7.4.1 Q
E
(ODD): EUCLIDEAN SELF-DUAL CODES OVER
Q
207
CLIFFORD-WEIL GROUPS (Q ODD) 207
THE CASE Q = 3 209
THE CASE Q = 3, GENUS 2 210
THE CASE Q = 9 211
THE CASE Q = 5 212
7.5 SYMPLECTIC TYPE, P ODD 213
7.5.1 G
H+
(ODD): HERMITIAN F
R
-LINEAR CODES OVER
Q
, Q = R
2
. . 214
CLIFFORD-WEIL GROUPS (GENUS G) 214
THE CASE Q = 9, GENUS 1 215
7.6 CHARACTERISTIC 2, ORTHOGONAL AND SYMPLECTIC TYPES 215
7.6.1 Q
U+
(EVEN): HERMITIAN F
R
-LINEAR CODES OVER
Q
, Q = R
2
. . 217
CLIFFORD-WEIL GROUPS (GENUS G) 217
THE CASE Q = 4, GENUS 1 217
THE CASE Q = A, GENUS 2 219
THE CASE Q = 16 220
7.6.2 Q
E
(EVEN): EUCLIDEAN SELF-DUAL F
9
-LINEAR CODES 220
CLIFFORD-WEIL GROUPS (GENUS G) 220
THE CASE Q = 2 221
THE CASE Q = A 221
CONTENTS XXI
7.6.3 Q^
+
(EVEN): EVEN TRACE-HERMITIAN F
R
-LINEAR CODES 222
CLIFFORD-WEIL GROUPS (GENUS G) 222
THE CASE Q = 4, GENUS 1 223
7.6.4 Q^ (EVEN): GENERALIZED DOUBLY-EVEN CODES OVER
Q
224
CLIFFORD-WEIL GROUPS (GENUS G) 224
THE CASE K = 2, ARBITRARY GENUS 225
THE CASE K = F
4
, GENUS 1 225
THE CASE K = F
8
226
FURTHER EXAMPLES OF SELF-DUAL CODES 227
8.1 M
Z
: CODES OVER Z/RNZ 227
8.2 4
Z
: SELF-DUAL CODES OVER Z/4Z 230
8.2.1 4
Z
: TYPE I SELF-DUAL CODES OVER Z/4Z 230
8.2.2 4
Z
: TYPE I SELF-DUAL CODES OVER Z/4Z CONTAINING 1 231
8.2.3 SAME, WITH 1 IN THE SHADOW 233
8.2.4 4F
T
: TYPE II SELF-DUAL CODES OVER Z/4Z 233
8.2.5 4F
U
: TYPE II SELF-DUAL CODES OVER Z/4Z CONTAINING 1 . 234
8.3 8
Z
: SELF-DUAL CODES OVER Z/8Z 234
8.4 CODES OVER MORE GENERAL GALOIS RINGS 235
8.4.1 GR(P
E
, /)
E
: EUCLIDEAN SELF-DUAL GR(P
E
, /)-LINEAR CODES. . 236
8.4.2 GR(P
E
, /)
H
: HERMITIAN SELF-DUAL GR(P
E
, /)-LINEAR CODES. 238
8.4.3 GR(P
E
, 2Z)
H+
: TRACE-HERMITIAN GR(P
E
, I)-LINEAR CODES. . 239
8.4.4 CLIFFORD-WEIL GROUPS FOR GR(4,2) 239
8.5 SELF-DUAL CODES OVER F,
2
+
Q
2 U 243
LATTICES 249
9.1 LATTICES AND THETA SERIES 252
9.1.1 PRELIMINARY DEFINITIONS 252
9.1.2 MODULAR LATTICES AND ATKIN-LEHNER INVOLUTIONS 255
9.1.3 SHADOWS 260
9.1.4 JACOBI FORMS 261
9.1.5 SIEGEL THETA SERIES 261
JACOBI-SIEGEL THETA SERIES AND RIEMANN THETA FUNCTIONS 265
RIEMANN THETA FUNCTIONS WITH HARMONIC COEFFICIENTS . . . 268
9.1.6 HILBERT THETA SERIES 269
9.2 POSITIVE DEFINITE FORM R-ALGEBRAS 272
9.3 HALF-SPACES 274
9.4 FORM ORDERS AND LATTICES 276
9.5 EVEN AND ODD UNIMODULAR LATTICES 278
9.6 GLUING THEORY FOR CODES 280
9.7 GLUING THEORY FOR LATTICES 282
XXII CONTENTS
10 MAXIMAL ISOTROPIC CODES AND LATTICES 285
10.1 MAXIMAL ISOTROPIC CODES 286
10.2 MAXIMAL ISOTROPIC DOUBLY-EVEN BINARY CODES 290
10.3 MAXIMAL ISOTROPIC EVEN BINARY CODES 293
10.4 MAXIMAL ISOTROPIC TERNARY CODES 293
10.5 MAXIMAL ISOTROPIC ADDITIVE CODES OVER F4 298
10.6 MAXIMAL ISOTROPIC CODES OVER Z/4Z 298
10.7 MAXIMAL EVEN LATTICES 301
10.7.1 MAXIMAL EVEN LATTICES OF DETERMINANT 3
FC
304
10.7.2 MAXIMAL EVEN AND INTEGRAL LATTICES OF DETERMINANT 2
K
. . 306
11 EXTREMAL AND OPTIMAL CODES 313
11.1 UPPER BOUNDS 314
11.1.1 EXTREMAL WEIGHT ENUMERATORS AND THE LP BOUND 314
11.1.2 SELF-DUAL BINARY CODES, 2
N
AND 2
R
317
11.1.3 SOME OTHER TYPES 321
11.1.4 A NEW DEFINITION OF EXTREMALITY 324
11.1.5 ASYMPTOTIC UPPER BOUNDS 326
11.2 LOWER BOUNDS 328
11.3 TABLES OF EXTREMAL SELF-DUAL CODES 331
11.3.1 BINARY CODES 331
11.3.2 TYPE 3: TERNARY CODES 336
11.3.3 TYPES 4
E
AND 4F
T
: EUCLIDEAN SELF-DUAL CODES OVER F
4
338
11.3.4 TYPE 4
H
: HERMITIAN LINEAR SELF-DUAL CODES OVER F4 339
11.3.5 TYPES 4
H+
AND 4"
+
: TRACE-HERMITIAN CODES OVER F
4
. . . 340
11.3.6 TYPE 4
Z
: SELF-DUAL CODES OVER Z/4Z 342
11.3.7 OTHER TYPES 345
12 ENUMERATION OF SELF-DUAL CODES 347
12.1 THE MASS FORMULAE 347
12.2 ENUMERATION OF BINARY SELF-DUAL CODES 350
INTERRELATIONS BETWEEN TYPES 2I AND 2N 356
12.3 TYPE 3: TERNARY SELF-DUAL CODES 360
12.3.1 TYPES 4
E
AND 4F
X
: EUCLIDEAN SELF-DUAL CODES OVER F
4
363
12.4 TYPE 4
H
: HERMITIAN SELF-DUAL CODES OVER F4 363
12.5 TYPE 4
H+
: TRACE-HERMITIAN ADDITIVE CODES OVER F
4
365
12.6 TYPE 4
Z
: SELF-DUAL CODES OVER Z/4Z 366
12.7 OTHER ENUMERATIONS 367
13 QUANTUM CODES 369
13.1 DEFINITIONS 370
13.2 ADDITIVE AND SYMPLECTIC QUANTUM CODES 373
13.3 HAMMING WEIGHT ENUMERATORS 376
CONTENTS XXIII
13.4 LINEAR PROGRAMMING BOUNDS 381
13.5 OTHER ALPHABETS 382
13.6 A TABLE OF QUANTUM CODES 385
REFERENCES 391
INDEX 417 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Nebe, Gabriele 1967- Rains, Eric M. Sloane, Neil J. A. 1939- |
| author_GND | (DE-588)17285671X (DE-588)121291553 |
| author_facet | Nebe, Gabriele 1967- Rains, Eric M. Sloane, Neil J. A. 1939- |
| author_role | aut aut aut |
| author_sort | Nebe, Gabriele 1967- |
| author_variant | g n gn e m r em emr n j a s nja njas |
| building | Verbundindex |
| bvnumber | BV021488291 |
| classification_rvk | SK 170 SK 950 SK 955 SK 260 SK 880 |
| classification_tum | DAT 580f |
| ctrlnum | (OCoLC)181523614 (DE-599)BVBBV021488291 |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV021488291 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:11:55Z |
| indexdate | 2024-07-09T20:36:56Z |
| institution | BVB |
| isbn | 9783540307297 354030729X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014705156 |
| oclc_num | 181523614 |
| open_access_boolean | |
| owner | DE-703 DE-91G DE-BY-TUM DE-634 DE-11 DE-188 DE-20 |
| owner_facet | DE-703 DE-91G DE-BY-TUM DE-634 DE-11 DE-188 DE-20 |
| physical | XXVI, 430 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series | Algorithms and computation in mathematics |
| series2 | Algorithms and computation in mathematics |
| spelling | Nebe, Gabriele 1967- Verfasser (DE-588)17285671X aut Self-dual codes and invariant theory with 34 tables Gabriele Nebe ; Eric M. Rains ; Neil J. A. Sloane Berlin [u.a.] Springer 2006 XXVI, 430 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algorithms and computation in mathematics 17 Auch als Internetausgabe Coding theory Configurações combinatórias. larpcal Duality theory (Mathematics) Invariants Teoria dos códigos. larpcal Selbstdualität (DE-588)4180824-1 gnd rswk-swf Invariantentheorie (DE-588)4162209-1 gnd rswk-swf Code (DE-588)4010345-6 gnd rswk-swf Code (DE-588)4010345-6 s Selbstdualität (DE-588)4180824-1 s Invariantentheorie (DE-588)4162209-1 s DE-604 Rains, Eric M. Verfasser aut Sloane, Neil J. A. 1939- Verfasser (DE-588)121291553 aut Algorithms and computation in mathematics 17 (DE-604)BV011131286 17 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2749052&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014705156&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nebe, Gabriele 1967- Rains, Eric M. Sloane, Neil J. A. 1939- Self-dual codes and invariant theory with 34 tables Algorithms and computation in mathematics Coding theory Configurações combinatórias. larpcal Duality theory (Mathematics) Invariants Teoria dos códigos. larpcal Selbstdualität (DE-588)4180824-1 gnd Invariantentheorie (DE-588)4162209-1 gnd Code (DE-588)4010345-6 gnd |
| subject_GND | (DE-588)4180824-1 (DE-588)4162209-1 (DE-588)4010345-6 |
| title | Self-dual codes and invariant theory with 34 tables |
| title_auth | Self-dual codes and invariant theory with 34 tables |
| title_exact_search | Self-dual codes and invariant theory with 34 tables |
| title_exact_search_txtP | Self-dual codes and invariant theory with 34 tables |
| title_full | Self-dual codes and invariant theory with 34 tables Gabriele Nebe ; Eric M. Rains ; Neil J. A. Sloane |
| title_fullStr | Self-dual codes and invariant theory with 34 tables Gabriele Nebe ; Eric M. Rains ; Neil J. A. Sloane |
| title_full_unstemmed | Self-dual codes and invariant theory with 34 tables Gabriele Nebe ; Eric M. Rains ; Neil J. A. Sloane |
| title_short | Self-dual codes and invariant theory |
| title_sort | self dual codes and invariant theory with 34 tables |
| title_sub | with 34 tables |
| topic | Coding theory Configurações combinatórias. larpcal Duality theory (Mathematics) Invariants Teoria dos códigos. larpcal Selbstdualität (DE-588)4180824-1 gnd Invariantentheorie (DE-588)4162209-1 gnd Code (DE-588)4010345-6 gnd |
| topic_facet | Coding theory Configurações combinatórias. Duality theory (Mathematics) Invariants Teoria dos códigos. Selbstdualität Invariantentheorie Code |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2749052&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014705156&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011131286 |
| work_keys_str_mv | AT nebegabriele selfdualcodesandinvarianttheorywith34tables AT rainsericm selfdualcodesandinvarianttheorywith34tables AT sloaneneilja selfdualcodesandinvarianttheorywith34tables |