Best Known (60−15, 60, s)-Nets in Base 3
(60−15, 60, 328)-Net over F3 — Constructive and digital
Digital (45, 60, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 15, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
(60−15, 60, 404)-Net over F3 — Digital
Digital (45, 60, 404)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(360, 404, F3, 15) (dual of [404, 344, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(360, 728, F3, 15) (dual of [728, 668, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(360, 728, F3, 15) (dual of [728, 668, 16]-code), using
(60−15, 60, 17749)-Net in Base 3 — Upper bound on s
There is no (45, 60, 17750)-net in base 3, because
- 1 times m-reduction [i] would yield (45, 59, 17750)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 14134 512099 704103 465357 392601 > 359 [i]