Best Known (4, 4+6, s)-Nets in Base 8
(4, 4+6, 25)-Net over F8 — Constructive and digital
Digital (4, 10, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
(4, 4+6, 33)-Net in Base 8 — Constructive
(4, 10, 33)-net in base 8, using
- base change [i] based on digital (0, 6, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
(4, 4+6, 37)-Net over F8 — Digital
Digital (4, 10, 37)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(810, 37, F8, 2, 6) (dual of [(37, 2), 64, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- a “GraX†code from Grassl’s database [i]
- OOA 2-folding [i] based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
(4, 4+6, 264)-Net in Base 8 — Upper bound on s
There is no (4, 10, 265)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1081 110696 > 810 [i]
- extracting embedded orthogonal array [i] would yield OA(810, 265, S8, 6), but
- the linear programming bound shows that M ≥ 59 925125 398528 / 55515 > 810 [i]