Best Known (88−11, 88, s)-Nets in Base 8
(88−11, 88, 3355454)-Net over F8 — Constructive and digital
Digital (77, 88, 3355454)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 14)-net over F8, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 1 and N(F) ≥ 14, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- digital (71, 82, 3355440)-net over F8, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- digital (1, 6, 14)-net over F8, using
(88−11, 88, large)-Net over F8 — Digital
Digital (77, 88, large)-net over F8, using
- t-expansion [i] based on digital (76, 88, large)-net over F8, using
- 1 times m-reduction [i] based on digital (76, 89, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(889, large, F8, 13) (dual of [large, large−89, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(889, large, F8, 13) (dual of [large, large−89, 14]-code), using
- 1 times m-reduction [i] based on digital (76, 89, large)-net over F8, using
(88−11, 88, large)-Net in Base 8 — Upper bound on s
There is no (77, 88, large)-net in base 8, because
- 9 times m-reduction [i] would yield (77, 79, large)-net in base 8, but