Best Known (87, 87+12, s)-Nets in Base 8
(87, 87+12, 2796224)-Net over F8 — Constructive and digital
Digital (87, 99, 2796224)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- digital (78, 90, 2796200)-net over F8, using
- net defined by OOA [i] based on linear OOA(890, 2796200, F8, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(890, 8388601, F8, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(890, 8388602, F8, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- trace code [i] based on linear OOA(6445, 4194301, F64, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6445, 8388602, F64, 12) (dual of [8388602, 8388557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- OOA 2-folding [i] based on linear OA(6445, 8388602, F64, 12) (dual of [8388602, 8388557, 13]-code), using
- trace code [i] based on linear OOA(6445, 4194301, F64, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(890, 8388602, F8, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(890, 8388601, F8, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(890, 2796200, F8, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- digital (3, 9, 24)-net over F8, using
(87, 87+12, large)-Net over F8 — Digital
Digital (87, 99, large)-net over F8, using
- 82 times duplication [i] based on digital (85, 97, large)-net over F8, using
- t-expansion [i] based on digital (83, 97, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(897, large, F8, 14) (dual of [large, large−97, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(897, large, F8, 14) (dual of [large, large−97, 15]-code), using
- t-expansion [i] based on digital (83, 97, large)-net over F8, using
(87, 87+12, large)-Net in Base 8 — Upper bound on s
There is no (87, 99, large)-net in base 8, because
- 10 times m-reduction [i] would yield (87, 89, large)-net in base 8, but