Best Known (78−20, 78, s)-Nets in Base 8
(78−20, 78, 819)-Net over F8 — Constructive and digital
Digital (58, 78, 819)-net over F8, using
- net defined by OOA [i] based on linear OOA(878, 819, F8, 20, 20) (dual of [(819, 20), 16302, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(878, 8190, F8, 20) (dual of [8190, 8112, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(878, 8192, F8, 20) (dual of [8192, 8114, 21]-code), using
- trace code [i] based on linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- trace code [i] based on linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(878, 8192, F8, 20) (dual of [8192, 8114, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(878, 8190, F8, 20) (dual of [8190, 8112, 21]-code), using
(78−20, 78, 7864)-Net over F8 — Digital
Digital (58, 78, 7864)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(878, 7864, F8, 20) (dual of [7864, 7786, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(878, 8192, F8, 20) (dual of [8192, 8114, 21]-code), using
- trace code [i] based on linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- trace code [i] based on linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(878, 8192, F8, 20) (dual of [8192, 8114, 21]-code), using
(78−20, 78, 7161101)-Net in Base 8 — Upper bound on s
There is no (58, 78, 7161102)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 27607 021894 154289 870350 478333 520029 474477 145299 849330 022424 430871 190734 > 878 [i]