Best Known (70−18, 70, s)-Nets in Base 8
(70−18, 70, 910)-Net over F8 — Constructive and digital
Digital (52, 70, 910)-net over F8, using
- net defined by OOA [i] based on linear OOA(870, 910, F8, 18, 18) (dual of [(910, 18), 16310, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(870, 8190, F8, 18) (dual of [8190, 8120, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(870, 8192, F8, 18) (dual of [8192, 8122, 19]-code), using
- trace code [i] based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- trace code [i] based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(870, 8192, F8, 18) (dual of [8192, 8122, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(870, 8190, F8, 18) (dual of [8190, 8120, 19]-code), using
(70−18, 70, 7612)-Net over F8 — Digital
Digital (52, 70, 7612)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(870, 7612, F8, 18) (dual of [7612, 7542, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(870, 8192, F8, 18) (dual of [8192, 8122, 19]-code), using
- trace code [i] based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- trace code [i] based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(870, 8192, F8, 18) (dual of [8192, 8122, 19]-code), using
(70−18, 70, 6261613)-Net in Base 8 — Upper bound on s
There is no (52, 70, 6261614)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1645 505177 706504 055220 251534 379664 969991 636613 101739 933146 989321 > 870 [i]