Best Known (210−45, 210, s)-Nets in Base 3
(210−45, 210, 688)-Net over F3 — Constructive and digital
Digital (165, 210, 688)-net over F3, using
- 32 times duplication [i] based on digital (163, 208, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 52, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 52, 172)-net over F81, using
(210−45, 210, 1720)-Net over F3 — Digital
Digital (165, 210, 1720)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3210, 1720, F3, 45) (dual of [1720, 1510, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 2186, F3, 45) (dual of [2186, 1976, 46]-code), using
- 1 times truncation [i] based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
- an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- 1 times truncation [i] based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 2186, F3, 45) (dual of [2186, 1976, 46]-code), using
(210−45, 210, 154317)-Net in Base 3 — Upper bound on s
There is no (165, 210, 154318)-net in base 3, because
- 1 times m-reduction [i] would yield (165, 209, 154318)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5228 645429 381763 492083 289336 317140 750252 467051 340031 236849 554857 206065 831659 936508 836264 527726 678821 > 3209 [i]