Best Known (140−35, 140, s)-Nets in Base 5
(140−35, 140, 416)-Net over F5 — Constructive and digital
Digital (105, 140, 416)-net over F5, using
- trace code for nets [i] based on digital (35, 70, 208)-net over F25, using
- net from sequence [i] based on digital (35, 207)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 35 and N(F) ≥ 208, using
- net from sequence [i] based on digital (35, 207)-sequence over F25, using
(140−35, 140, 2870)-Net over F5 — Digital
Digital (105, 140, 2870)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5140, 2870, F5, 35) (dual of [2870, 2730, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(5140, 3124, F5, 35) (dual of [3124, 2984, 36]-code), using
- 1 times truncation [i] based on linear OA(5141, 3125, F5, 36) (dual of [3125, 2984, 37]-code), using
- an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 1 times truncation [i] based on linear OA(5141, 3125, F5, 36) (dual of [3125, 2984, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(5140, 3124, F5, 35) (dual of [3124, 2984, 36]-code), using
(140−35, 140, 931080)-Net in Base 5 — Upper bound on s
There is no (105, 140, 931081)-net in base 5, because
- 1 times m-reduction [i] would yield (105, 139, 931081)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 14 349453 681890 869611 986869 084733 223930 540970 889804 049130 997071 548876 098225 148747 846821 682066 500325 > 5139 [i]