Best Known (38, 38+9, s)-Nets in Base 8
(38, 38+9, 65544)-Net over F8 — Constructive and digital
Digital (38, 47, 65544)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (34, 43, 65535)-net over F8, using
- net defined by OOA [i] based on linear OOA(843, 65535, F8, 9, 9) (dual of [(65535, 9), 589772, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(843, 262141, F8, 9) (dual of [262141, 262098, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(843, 262141, F8, 9) (dual of [262141, 262098, 10]-code), using
- net defined by OOA [i] based on linear OOA(843, 65535, F8, 9, 9) (dual of [(65535, 9), 589772, 10]-NRT-code), using
- digital (0, 4, 9)-net over F8, using
(38, 38+9, 262166)-Net over F8 — Digital
Digital (38, 47, 262166)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(847, 262166, F8, 9) (dual of [262166, 262119, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(825, 262144, F8, 5) (dual of [262144, 262119, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(84, 22, F8, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,8)), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
(38, 38+9, large)-Net in Base 8 — Upper bound on s
There is no (38, 47, large)-net in base 8, because
- 7 times m-reduction [i] would yield (38, 40, large)-net in base 8, but