6.7 New Codes from Old
161
Table 6.5 (continued)
Code Family / [[n, k, d z /d x ]] q
Range of Parameters
Ref.
Expanded Character
[[t2 m , t[k(r 2 ) − k(r 1 )], d z /d x ]] p
q = p t , p odd prime, t ≥ 1,
k(r ) =
r
i=0
m
i
, d z ≥ 2 m−r 2 ,
d x ≥ 2 r 1 +1
QR
[[ p, 1, d z /d x ]] q
p prime, p ≡ 1 mod 4,
[80]
q = p t
1 , p p 1 , q is a quadratic
residue mod p,
d z ≥
√ p, d x ≥
√ p
[[ p, 1, d z /d x ]] q
p prime, p ≡ 3 mod 4,
[80]
q = p t
1 , p p 1 , q is a quadratic
residue mod p,
d z ≥ d, d x ≥ d, d 2 − d + 1 ≥ p
Expanded QR
[[tp, t, d z /d x ]] p∗
p prime, p ≡ 1 mod 4, q = p t
∗ ,
t ≥ 1, p p ∗ , q is a quadratic
residue mod p,
d z ≥
√ p, d x ≥
√ p
[[tp, t, d z /d x ]] p∗
p prime, p ≡ 3 mod 4, q = p t
∗ ,
t ≥ 1, p p ∗ , q is a quadratic
residue mod p,
d z ≥ d, d x ≥ d, d 2 − d + 1 ≥ p
Affine-Invariant
[[tp m , t ( p m − 2 − 2 m
t ), d z /d x ]] p
q = p t ,
p > 3, m > 2, d z ≥ d a , d x ≥ d a ,
d a is given in Theorem 6.7.16
Product code
[[(q − 1) 2 , (q − d 1 )(q − d 3 ) − (q − d 2 )(q − d 4 ), d z /d x ]] q
2 ≤ d 1 ≤ d 2 < q − 1,
2 ≤ d 3 ≤ d 4 < q − 1,
[95]
d z ≥
max{d 1 d 3 , min{q − d 2 , q − d 4 }},
d x ≥
min{d 1 d 3 , min{q − d 2 , q − d 4 }}
6.7.6.6 Code Tables
Here, we show Tables 6.4 and 6.5 containing families of AQECCs available in the
literature as well as the code families constructed in this chapter. In the first column,
we exhibit the class and the parameters [[n, k, d z /d x ]] q of an AQECC; in the second
column the parameter’s range and in the third column, the corresponding references
are shown.