160
6 Asymmetric Quantum Codes
Theorem 6.7.16 Assume that q = p
t , m is a positive integer and n = p
m
− 1. If p >
3 or m > 2 or t = 1, then there exists an [[t p
m
, t ( p
m
− 2 − 2
m
t
), d z /d x ]] p AQECC,
where d z ≥ d a , d x ≥ d a , and d a is the minimum distance of an extended maximal
affine-invariant code.
Proof Consider the dual-containing extended maximal affine-invariant code C
e with
parameters [ p
m
, p
m
− 1 − m/t, d] given in Lemma 6.7.2, where p > 3 (or m > 2
or t = 1). Applying Theorem 6.7.3 we have an [[t p
m
, t ( p
m
− 2 − 2
m
t
), d z /d x ]] p
AQECC, where d z ≥ d a , d x ≥ d a and d a is the minimum distance of C
e .
Table 6.5 Families of AQECCs
Code Family / [[n, k, d z /d x ]] q
Range of Parameters
Ref.
RM
[[2 m , k, 2 m−r 2 ≥ 2 r 1 +1 ]] 2
0 ≤ r 1 < r 2 < m,
k =
r 2
j=r 1 +1
m
j
[140]
Expanded GRM
[[lq m , l[k(α 2 ) − k(α 1 )], d z /d x ]] p
0 ≤ α 1 ≤ α 2 < m(q − 1), q = p l ,
p prime, l ≥ 1,
k(α) =
m
i=0
(−1) i
m
i
m + α − iq
α − iq
,
d z ≥ d(α 2 ), d x ≥ d(α ⊥
1 ),
d(α 2 ) = (t + 1)q u ,
m(q − 1) − α 2 = (q − 1)u + t,
0 ≤ t < q − 1,
d(α ⊥
1 ) = (a + 1)q b ,
α 1 + 1 = (q − 1)b + a,
0 ≤ a ≤ q − 1
MDS
[[n, n − d 1 − d 2 + 2, d z /d x ]] q
n = q − 1, d x = d 1 < d z = d 2
[3]
[[n, n − 2, 2/2]] q
q prime power, n ≥ 3
[157]
[[n, k − 1, (n − k + 1)/2]] q
q ≥ n > 3, 1 < k ≤ n − 2
[157]
[[2 m + 2, 2, 2 m /2]] 2 m
m > 0 integer
[157]
[[2 m + 2, 2 m − 2, 4/2]] 2 m
m > 0, m = 2 integer
[157]
[[n, j, d z /d x ]] q
n, k, j ∈ Z, q ≥ 5, n ≤ q,
2 ≤ k ≤ n − 3,
[157]
j ≤ n − k − 2,
{d z , d x } = {n − k − j + 1, k + 1}
[[q + 1, 2 j, d z /d x ]] q
n, k, j ∈ Z, q ≥ 5, k ≥ 2,
k + 2 j ≤ q − 1,
[157]
{d z , d x } = {q − k − 2 j + 2, k + 1}
[[q + 1, q − 1 − 2s, (2s + 1)/3]] q
q = 2 m ≥ 4, s ≤ q/2 − 1
[157]
[[2 m + 2, 2 m − 4, 4/4]] 2 m
2 m ≥ 4
[157]
[[n, 2k − n + c, d z ≥ d/d x ≥ (d − c)]] q
1 < k < n < 2k + c ≤ q,
[94]
k = n − d + 1, d > c + 1, c ≥ 1
(continued)