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5 Quantum Code Constructions
and w
(3)
j ∈ C 3 . In other words, each c ∈ C 1 ⊗ C 3 can be written as c =
i
v
(1)
i ⊗
w
(3)
j . An (Euclidean) inner product on C 1 ⊗ C 3 can be defined as
v
(1)
i ⊗ w
(3)
i |v
(1)
j ⊗ w
(3)
j
= =v
(1)
i |v
(1)
j w
(3)
i |w
(3)
j ,
(5.2)
and it is extended by linearity for all elements of C 1 ⊗ C 3 . Note that c
(1)
i |c
(1)
j and
c
(3)
i |c
(3)
j are the Euclidean inner products on C 1 and C 3 , respectively. Since C 1 is
self-orthogonal, it follows from Eq. 5.2 that C 1 ⊗ C 3 is also self-orthogonal, which
implies that (C 1 ⊗ C 3 )
⊥ is dual-containing. The parameters of the codes (C 1 ⊗ C 3 )
⊥
and C 2 ⊗ C 4 are [nn
∗
, nn
∗
− k 1 k 3 , d] and [nn
∗
, k 2 k 2 , d 2 d 4 ], respectively. Since
(C 1 ⊗ C 3 )
⊥
C 2 ⊗ C 4 , it follows that d ≥ d 2 d 4 . Applying Theorem 5.5.1 to the
cyclic codes (C 1 ⊗ C 3 )
⊥ and C 2 ⊗ C 4 the proof is complete.
5 Quantum Code Constructions
and w
(3)
j ∈ C 3 . In other words, each c ∈ C 1 ⊗ C 3 can be written as c =
i
v
(1)
i ⊗
w
(3)
j . An (Euclidean) inner product on C 1 ⊗ C 3 can be defined as
v
(1)
i ⊗ w
(3)
i |v
(1)
j ⊗ w
(3)
j
= =v
(1)
i |v
(1)
j w
(3)
i |w
(3)
j ,
(5.2)
and it is extended by linearity for all elements of C 1 ⊗ C 3 . Note that c
(1)
i |c
(1)
j and
c
(3)
i |c
(3)
j are the Euclidean inner products on C 1 and C 3 , respectively. Since C 1 is
self-orthogonal, it follows from Eq. 5.2 that C 1 ⊗ C 3 is also self-orthogonal, which
implies that (C 1 ⊗ C 3 )
⊥ is dual-containing. The parameters of the codes (C 1 ⊗ C 3 )
⊥
and C 2 ⊗ C 4 are [nn
∗
, nn
∗
− k 1 k 3 , d] and [nn
∗
, k 2 k 2 , d 2 d 4 ], respectively. Since
(C 1 ⊗ C 3 )
⊥
C 2 ⊗ C 4 , it follows that d ≥ d 2 d 4 . Applying Theorem 5.5.1 to the
cyclic codes (C 1 ⊗ C 3 )
⊥ and C 2 ⊗ C 4 the proof is complete.