Best Known (32, 32+13, s)-Nets in Base 3
(32, 32+13, 144)-Net over F3 — Constructive and digital
Digital (32, 45, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 15, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
(32, 32+13, 190)-Net over F3 — Digital
Digital (32, 45, 190)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(345, 190, F3, 13) (dual of [190, 145, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(345, 242, F3, 13) (dual of [242, 197, 14]-code), using
- the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(345, 242, F3, 13) (dual of [242, 197, 14]-code), using
(32, 32+13, 4716)-Net in Base 3 — Upper bound on s
There is no (32, 45, 4717)-net in base 3, because
- 1 times m-reduction [i] would yield (32, 44, 4717)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 986 001993 534276 432465 > 344 [i]