6.7 New Codes from Old
149
Theorem 6.7.3 Let q = p
t be a prime power. If there exists an ((n, K , d z /d x )) q m
stabilizer code, then there exists an ((nm, K , d
∗
z /d
∗
x )) q stabilizer code, where d
∗
z ≥ d z
and d
∗
x ≥ d x .
Proof If a is an element of F q m , we can expand a with respect to a given basis
B = {β 1 , . . . , β m } of F q m over F q and put the coordinates of a in the vector form
c B (a) = (a 1 , . . . , a m ) ∈ F
m
q . Let us consider the non-degenerate symmetric form
tr q m /q (ab) on the vector space F q m (over F q ). Assume that ϕ B is the F p -vector
space isomorphism from F
2n
q m to F
2nm
q
given (in the proof of [80, Lemma 76]) by
ϕ B ((u|v)) = ((c B (u 1 ), . . . , c B (u n ))|(Mc B (v 1 ), . . . , Mc B (v n ))), where u, v ∈ F
n
q m
are given by u = (u 1 , . . . , u n ) and v = (v 1 , . . . , v n ), M = (tr q m /q (β i β j )) 1≤i, j≤m is
the Gram matrix and tr q m /q (ab) = c B (a)
t Mc B (b) for all a, b ∈ F q m . Note that the
inner product considered here is the usual Euclidean inner product of F q .
Assume that an ((n, K , d z /d x )) q m stabilizer code exists. From [80, Theorem 13],
there exists an additive code C ≤ F
2n
q m of size |C| = q
mn
/K such that C ≤ C
⊥ s ,
wt X (C
⊥ s \C) = d x if K > 1 (and wt X (C
⊥ s ) = d x if K = 1) and wt Z (C
⊥ s \C) = d z if
K > 1 (and wt Z (C
⊥ s ) = d z if K = 1). We know that ϕ B preserves trace-symplectic
orthogonality, that is, the code ϕ B (C) satisfies ϕ B (C) ≤ [ϕ B (C)]
⊥ s . Let (u|v) ∈ F
2n
q m
and u i = 0 (resp. v j = 0) for some i ∈ {1, . . . , n} (resp. j ∈ {1, . . . , n}). Hence, at
least one coordinate of the corresponding vector c B (u i ) (resp. Mc B (v j )) is nonzero.
Thus wt X ([ϕ B (C)]
⊥ s \ϕ B (C)) ≥ d x if K > 1 (and wt X ([ϕ B (C)]
⊥ s ) ≥ d x if K = 1)
and wt Z ([ϕ B (C)]
⊥ s \ϕ B (C)) ≥ d z if K > 1 (and wt Z ([ϕ B (C)]
⊥ s ) ≥ d z if K = 1).
Because the alphabet considered now is F q , then there exists an ((nm, K , d
∗
z /d
∗
x )) q
stabilizer code, where d
∗
z ≥ d z and d
∗
x ≥ d x . The proof is complete.
6.7.2 Direct Sum Codes
Let us recall the direct sum of codes (see Definition 4.3.6). Let C 1 = [n 1 , k 1 , d 1 ] q
and C 2 = [n 2 , k 2 , d 2 ] q be two linear codes. Then the direct sum code C 1 ⊕ C 2 is the
linear code given by
C 1 ⊕ C 2 = {(c 1 , c 2 )|c 1 ∈ C 1 , c 2 ∈ C 2 },
with parameters [n 1 + n 2 , k 1 + k 2 , min {d 1 , d 2 }] q .
In the sequence, we show how to obtain new AQECCs by applying direct sum of
codes.
Theorem 6.7.4 Let q be a prime power. Assume there exists an [[n, k, d z /d x ]] q
AQECC derived from two linear nested codes C 1 = [n, k 1 , d 1 ] q and C 2 = [n, k 2 , d 2 ] q
with C 2 ⊂ C 1 . Suppose also the existence of an [[n
∗
, k
∗
, d
∗
z /d
∗
x ]] q AQECC derived
from linear nested codes C 3 = [n
∗
, k 3 , d 3 ] q and C 4 = [n
∗
, k 4 , d 4 ] q with C 4 ⊂ C 3 .
Then there exists an [[n + n
∗
, k + k
∗
, d
z /d
x ]] q = [[n + n
∗
, (k 1 + k 3 ) − (k 2 + k 4 ),
d
z /d
x ]] q AQECC, where d
z ≥ min{d 1 , d 3 }, d
x ≥ min{d
⊥
2 , d
⊥
4 } and d
⊥
2 , d
⊥
4 are the
minimum distances of the dual codes C
⊥
2 and C
⊥
4 , respectively.