226
Index
Even-like vector, 152
Extended code, 152
F
Field, 216
Five qubit code, 33
Free distance of a convolutional code, 164
Free module, 8
G
Generalized Reed–Muller code, 154
Generalized Reed–Solomon code, 139
General Pauli group, 33
Generator matrix in standard form, 44
Generator matrix of a linear code, 43
Genus of a function field, 54
Group, 213
H
Hadamard gate, 21
Hamming distance, 44, 45
Hamming weight, 45
Hermitian construction, 67
Hermitian curve, 111
Hermitian dual code, 46
Hermitian inner product, 46
Hermitian operator, 12
Hilbert space, 10
Homomorphism of modules, 7
I
Ideal, 216
Image of a linear transformation, 4
Inner product, 10
Integral domain, 215
Isomorphism of vector space, 3
K
Kernel, 4
Kronecker product, 14
L
Linear codes, 43
Linear combination, 2
Linear congruence, 62
Linearly dependent vectors, 3
Linearly independent vectors, 3
Linear operator, 4, 5
Linear transformation, 3, 4
Local ring, 53
M
Matrix of a linear transformation, 4
Measurement operators, 17
Memory of a convolutional code, 164
Memory of a quantum convolutional code,
167
Metric space, 8
Minimal polynomial, 50
Module, 7
Multiplicative order of q modulo n, 50
N
Narrow-sense BCH code, 52
Negacyclic code, 186
Nice error basis, 35
No-cloning theorem, 25
Normal operator, 12
Normal subgroup, 214
Norm of a vector, 11
O
Odd-like vector, 152
Orthogonal basis, 10
Orthonormal basis, 10
Overall constraint length, 164
Overall constraint length of a quantum convolutional code, 166
P
Parameters of a linear code, 45
Parameters of a quantum code, 37
Parity check matrix of a linear code, 44
Pauli matrices, 11
Place, 53
Positive operator, 12
Postulates of quantum mechanics, 17
Primitive BCH code, 52
Principal divisor, 54
Projective measurement, 18
Projector, 13
Punctured code, 47
Pure state, 25
Q
Quadratic residue code, 157
Quantum bit (qubit), 11
Quantum convolutional code, 166, 167
Index
Even-like vector, 152
Extended code, 152
F
Field, 216
Five qubit code, 33
Free distance of a convolutional code, 164
Free module, 8
G
Generalized Reed–Muller code, 154
Generalized Reed–Solomon code, 139
General Pauli group, 33
Generator matrix in standard form, 44
Generator matrix of a linear code, 43
Genus of a function field, 54
Group, 213
H
Hadamard gate, 21
Hamming distance, 44, 45
Hamming weight, 45
Hermitian construction, 67
Hermitian curve, 111
Hermitian dual code, 46
Hermitian inner product, 46
Hermitian operator, 12
Hilbert space, 10
Homomorphism of modules, 7
I
Ideal, 216
Image of a linear transformation, 4
Inner product, 10
Integral domain, 215
Isomorphism of vector space, 3
K
Kernel, 4
Kronecker product, 14
L
Linear codes, 43
Linear combination, 2
Linear congruence, 62
Linearly dependent vectors, 3
Linearly independent vectors, 3
Linear operator, 4, 5
Linear transformation, 3, 4
Local ring, 53
M
Matrix of a linear transformation, 4
Measurement operators, 17
Memory of a convolutional code, 164
Memory of a quantum convolutional code,
167
Metric space, 8
Minimal polynomial, 50
Module, 7
Multiplicative order of q modulo n, 50
N
Narrow-sense BCH code, 52
Negacyclic code, 186
Nice error basis, 35
No-cloning theorem, 25
Normal operator, 12
Normal subgroup, 214
Norm of a vector, 11
O
Odd-like vector, 152
Orthogonal basis, 10
Orthonormal basis, 10
Overall constraint length, 164
Overall constraint length of a quantum convolutional code, 166
P
Parameters of a linear code, 45
Parameters of a quantum code, 37
Parity check matrix of a linear code, 44
Pauli matrices, 11
Place, 53
Positive operator, 12
Postulates of quantum mechanics, 17
Primitive BCH code, 52
Principal divisor, 54
Projective measurement, 18
Projector, 13
Punctured code, 47
Pure state, 25
Q
Quadratic residue code, 157
Quantum bit (qubit), 11
Quantum convolutional code, 166, 167