Best Known (37−17, 37, s)-Nets in Base 25
(37−17, 37, 156)-Net over F25 — Constructive and digital
Digital (20, 37, 156)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- digital (9, 26, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- digital (3, 11, 52)-net over F25, using
(37−17, 37, 599)-Net over F25 — Digital
Digital (20, 37, 599)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2537, 599, F25, 17) (dual of [599, 562, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2537, 639, F25, 17) (dual of [639, 602, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(2533, 625, F25, 17) (dual of [625, 592, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2523, 625, F25, 12) (dual of [625, 602, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(254, 14, F25, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(2537, 639, F25, 17) (dual of [639, 602, 18]-code), using
(37−17, 37, 306339)-Net in Base 25 — Upper bound on s
There is no (20, 37, 306340)-net in base 25, because
- 1 times m-reduction [i] would yield (20, 36, 306340)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 211 759327 839943 683466 543642 128628 936402 070728 442113 > 2536 [i]