Best Known (75−17, 75, s)-Nets in Base 3
(75−17, 75, 400)-Net over F3 — Constructive and digital
Digital (58, 75, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (58, 76, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 19, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 19, 100)-net over F81, using
(75−17, 75, 712)-Net over F3 — Digital
Digital (58, 75, 712)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(375, 712, F3, 17) (dual of [712, 637, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(375, 738, F3, 17) (dual of [738, 663, 18]-code), using
- (u, u+v)-construction [i] based on
- linear OA(38, 9, F3, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,3)), using
- dual of repetition code with length 9 [i]
- linear OA(367, 729, F3, 17) (dual of [729, 662, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(38, 9, F3, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(375, 738, F3, 17) (dual of [738, 663, 18]-code), using
(75−17, 75, 48748)-Net in Base 3 — Upper bound on s
There is no (58, 75, 48749)-net in base 3, because
- 1 times m-reduction [i] would yield (58, 74, 48749)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 202758 740214 842866 608426 975585 989985 > 374 [i]