Best Known (69−56, 69, s)-Nets in Base 5
(69−56, 69, 34)-Net over F5 — Constructive and digital
Digital (13, 69, 34)-net over F5, using
- net from sequence [i] based on digital (13, 33)-sequence over F5, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F5 with g(F) = 11, N(F) = 32, and 2 places with degree 2 [i] based on function field F/F5 with g(F) = 11 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(69−56, 69, 36)-Net over F5 — Digital
Digital (13, 69, 36)-net over F5, using
- net from sequence [i] based on digital (13, 35)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 13 and N(F) ≥ 36, using
(69−56, 69, 72)-Net over F5 — Upper bound on s (digital)
There is no digital (13, 69, 73)-net over F5, because
- 1 times m-reduction [i] would yield digital (13, 68, 73)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(568, 73, F5, 55) (dual of [73, 5, 56]-code), but
- residual code [i] would yield linear OA(513, 17, F5, 11) (dual of [17, 4, 12]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(568, 73, F5, 55) (dual of [73, 5, 56]-code), but
(69−56, 69, 74)-Net in Base 5 — Upper bound on s
There is no (13, 69, 75)-net in base 5, because
- 3 times m-reduction [i] would yield (13, 66, 75)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(566, 75, S5, 53), but
- the linear programming bound shows that M ≥ 4684 092198 316280 875047 823428 758420 050144 195556 640625 / 235521 > 566 [i]
- extracting embedded orthogonal array [i] would yield OA(566, 75, S5, 53), but