Best Known (8, 39, s)-Nets in Base 5
(8, 39, 23)-Net over F5 — Constructive and digital
Digital (8, 39, 23)-net over F5, using
- net from sequence [i] based on digital (8, 22)-sequence over F5, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F5 with g(F) = 7, N(F) = 22, and 1 place with degree 2 [i] based on function field F/F5 with g(F) = 7 and N(F) ≥ 22, using an explicitly constructive algebraic function field [i]
(8, 39, 60)-Net over F5 — Upper bound on s (digital)
There is no digital (8, 39, 61)-net over F5, because
- 1 times m-reduction [i] would yield digital (8, 38, 61)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(538, 61, F5, 30) (dual of [61, 23, 31]-code), but
- construction Y1 [i] would yield
- linear OA(537, 43, F5, 30) (dual of [43, 6, 31]-code), but
- residual code [i] would yield linear OA(57, 12, F5, 6) (dual of [12, 5, 7]-code), but
- OA(523, 61, S5, 18), but
- the linear programming bound shows that M ≥ 6321 833259 448409 889221 191406 250000 / 502952 782605 714137 > 523 [i]
- linear OA(537, 43, F5, 30) (dual of [43, 6, 31]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(538, 61, F5, 30) (dual of [61, 23, 31]-code), but
(8, 39, 63)-Net in Base 5 — Upper bound on s
There is no (8, 39, 64)-net in base 5, because
- extracting embedded orthogonal array [i] would yield OA(539, 64, S5, 31), but
- the linear programming bound shows that M ≥ 146538 716374 971525 969029 926272 924058 139324 188232 421875 / 78 556145 493236 090390 497512 > 539 [i]