Best Known (58, 58+18, s)-Nets in Base 8
(58, 58+18, 3641)-Net over F8 — Constructive and digital
Digital (58, 76, 3641)-net over F8, using
- net defined by OOA [i] based on linear OOA(876, 3641, F8, 18, 18) (dual of [(3641, 18), 65462, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(876, 32769, F8, 18) (dual of [32769, 32693, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(876, 32773, F8, 18) (dual of [32773, 32697, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(871, 32768, F8, 17) (dual of [32768, 32697, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(80, 5, F8, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(876, 32773, F8, 18) (dual of [32773, 32697, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(876, 32769, F8, 18) (dual of [32769, 32693, 19]-code), using
(58, 58+18, 16612)-Net over F8 — Digital
Digital (58, 76, 16612)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(876, 16612, F8, 18) (dual of [16612, 16536, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using
(58, 58+18, large)-Net in Base 8 — Upper bound on s
There is no (58, 76, large)-net in base 8, because
- 16 times m-reduction [i] would yield (58, 60, large)-net in base 8, but