Best Known (109, 109+40, s)-Nets in Base 5
(109, 109+40, 408)-Net over F5 — Constructive and digital
Digital (109, 149, 408)-net over F5, using
- t-expansion [i] based on digital (105, 149, 408)-net over F5, using
- 1 times m-reduction [i] based on digital (105, 150, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 75, 204)-net over F25, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 30 and N(F) ≥ 204, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- trace code for nets [i] based on digital (30, 75, 204)-net over F25, using
- 1 times m-reduction [i] based on digital (105, 150, 408)-net over F5, using
(109, 109+40, 1822)-Net over F5 — Digital
Digital (109, 149, 1822)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5149, 1822, F5, 40) (dual of [1822, 1673, 41]-code), using
- 1672 step Varšamov–Edel lengthening with (ri) = (9, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 36 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 44 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 52 times 0, 1, 55 times 0, 1, 58 times 0, 1, 59 times 0, 1, 63 times 0, 1, 65 times 0, 1, 68 times 0, 1, 71 times 0) [i] based on linear OA(540, 41, F5, 40) (dual of [41, 1, 41]-code or 41-arc in PG(39,5)), using
- dual of repetition code with length 41 [i]
- 1672 step Varšamov–Edel lengthening with (ri) = (9, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 36 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 44 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 52 times 0, 1, 55 times 0, 1, 58 times 0, 1, 59 times 0, 1, 63 times 0, 1, 65 times 0, 1, 68 times 0, 1, 71 times 0) [i] based on linear OA(540, 41, F5, 40) (dual of [41, 1, 41]-code or 41-arc in PG(39,5)), using
(109, 109+40, 334621)-Net in Base 5 — Upper bound on s
There is no (109, 149, 334622)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 140 133312 811781 911901 462610 454211 488903 499342 137765 546990 816047 163533 041160 968796 081906 288904 291273 945921 > 5149 [i]