Best Known (85, 85+157, s)-Nets in Base 3
(85, 85+157, 60)-Net over F3 — Constructive and digital
Digital (85, 242, 60)-net over F3, using
- net from sequence [i] based on digital (85, 59)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 59)-sequence over F9, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 59)-sequence over F9, using
(85, 85+157, 84)-Net over F3 — Digital
Digital (85, 242, 84)-net over F3, using
- t-expansion [i] based on digital (71, 242, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(85, 85+157, 295)-Net over F3 — Upper bound on s (digital)
There is no digital (85, 242, 296)-net over F3, because
- 1 times m-reduction [i] would yield digital (85, 241, 296)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3241, 296, F3, 156) (dual of [296, 55, 157]-code), but
- residual code [i] would yield OA(385, 139, S3, 52), but
- the linear programming bound shows that M ≥ 25187 471495 546450 938829 166628 945306 291038 411946 137194 049987 820699 630759 367931 165327 613809 362902 650627 817783 270689 253835 813240 163063 / 655379 887178 349656 963856 190956 437056 740438 060960 854359 735199 515832 284477 226278 150220 800000 > 385 [i]
- residual code [i] would yield OA(385, 139, S3, 52), but
- extracting embedded orthogonal array [i] would yield linear OA(3241, 296, F3, 156) (dual of [296, 55, 157]-code), but
(85, 85+157, 372)-Net in Base 3 — Upper bound on s
There is no (85, 242, 373)-net in base 3, because
- 1 times m-reduction [i] would yield (85, 241, 373)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10 622811 848883 298425 665769 733685 271761 275918 969642 744358 720623 354736 501335 502506 463963 840043 471699 583220 717633 347521 > 3241 [i]