6.7 New Codes from Old
151
Theorem 6.7.6 Assume there exists an [[n, k, d z /d x ]] q stabilizer code derived from
two linear codes C 1 = [n, k 1 , d 1 ] q and C 2 = [n, k 2 , d 2 ] q , with C 2 ⊂ C 1 , n ≥ 2, k =
k 1 − k 2 , d z ≥ d 1 and d x ≥ d
⊥
2 , where d
⊥
2 is the minimum distance of the dual code
C
⊥
2 . Suppose also that d 1 ≥ 2, d
⊥
2 ≥ 2, and assume that C
⊥
2 contains at least a
nonzero codeword with ith coordinate zero. Then the following hold:
(i) If C 1 has a minimum weight codeword with a nonzero ith coordinate, then there
exists an [[n − 1, k, d
P i
z /d
P i
x ]] q AQECC, where k = k 1 − k 2 , d
P i
z ≥ d 1 − 1 and
d
P i
x ≥ d
⊥
2 ;
(ii) If C 1 has no minimum weight codeword with a nonzero ith coordinate, then
there exists an [[n − 1, k, d
P i
z /d
P i
x ]] q AQECC, where k = k 1 − k 2 , d
P i
z ≥ d 1 and
d
P i
x ≥ d
⊥
2 ≥ 2.
Proof We only prove Item (ii) since the proof of Item (i) is similar to this one.
Consider the punctured codes C
P i
1 and C
P i
2 . Since the inclusion C 2 ⊂ C 1 holds, it
follows that C
P i
2 ⊂ C
P i
1 . Since from hypothesis one has d 1 > 1 then it follows that
d 2 > 1 because C 2 ⊂ C 1 ; again from the hypothesis C 1 has no minimum weight
codeword with a nonzero ith coordinate. Thus, by Theorem [67, Theorem 1.5.1], the
punctured codes C
P i
1 and C
P i
2 have parameters [n − 1, k 1 , d 1 ] q and [n − 1, k 2 , d
i
2 ] q ,
respectively, where d
i
2 = d 2 or d
i
2 = d 2 − 1.
We need to compute the minimum distance of [C
P i
2 ]
⊥ in order to apply the CSS
construction. To do this, let us consider the code [C
P i
2 ]
⊥ . Since C
⊥
2 contains at least
a nonzero codeword whose ith coordinate is equal to zero, it follows that C
⊥
2 has a
subcode C
⊥
2 ({i}) = {0} and, consequently, the minimum distance d (C
⊥
2 ) i
of C
⊥
2 ({i})
satisfies d (C
⊥
2 ) i
≥ d
⊥
2 , where d
⊥
2 > 1. Since d (C
⊥
2 ) i
> 1, and from definition, the code
C
⊥
2 ({i}) has no minimum weight codeword with a nonzero ith coordinate. Applying
again Theorem [67, Theorem 1.5.1], we conclude that the shortened code [C
⊥
2 ] S i has
minimum distance d (C
⊥
2 ) i
. From [67, Theorem 1.5.7], we know that [C
P i
2 ]
⊥ = [C
⊥
2 ] S i ;
hence, the code [C
P i
2 ]
⊥ has minimum distance d (C
⊥
2 ) i
, where d (C
⊥
2 ) i
≥ d
⊥
2 . Therefore,
applying the CSS construction to the codes C
P i
1 , C
P i
2 and [C
P i
2 ]
⊥ , one can derive
an [[n − 1, k, d
P i
z /d
P i
x ]] q AQECC, where k = k 1 − k 2 , d
P i
z ≥ d 1 and d
P i
x ≥ d (C
⊥
2 ) i
≥
d
⊥
2 ≥ 2. The proof is complete.
Following the lines adopted in [80], we can show a more general result.
Theorem 6.7.7 Assume that a pure [[n, k, d z /d x ]] q stabilizer code exists, where
n ≥ 2 and d x , d z ≥ 2. Then there exists a pure [[n − 1, k, d
∗
z /d
∗
x ]] q stabilizer code,
where d
∗
z ≥ d z − 1 and d
∗
x ≥ d x − 1.
Proof Assume that a pure [[n, k, d z /d x ]] q stabilizer code with minimum distance
d exists. From [80, Corollary 72], there exists a pure [[n − 1, k, d
∗
≥ d − 1]] q stabilizer code derived from an additive self-orthogonal (with respect to the tracealternating form) code D
⊥ a ≤ F
n−1
q 2 with wt(D
⊥ a ) ≥ d − 1. Let us consider the vectors v, w ∈ F
2(n−1)
q
. Let (β, β
q
) be a normal basis of F
2
q over F q . We know that the