Best Known (165, 165+33, s)-Nets in Base 3
(165, 165+33, 1480)-Net over F3 — Constructive and digital
Digital (165, 198, 1480)-net over F3, using
- 32 times duplication [i] based on digital (163, 196, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 49, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 49, 370)-net over F81, using
(165, 165+33, 7859)-Net over F3 — Digital
Digital (165, 198, 7859)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3198, 7859, F3, 2, 33) (dual of [(7859, 2), 15520, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3198, 9841, F3, 2, 33) (dual of [(9841, 2), 19484, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3198, 19682, F3, 33) (dual of [19682, 19484, 34]-code), using
- 1 times truncation [i] based on linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 1 times truncation [i] based on linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using
- OOA 2-folding [i] based on linear OA(3198, 19682, F3, 33) (dual of [19682, 19484, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(3198, 9841, F3, 2, 33) (dual of [(9841, 2), 19484, 34]-NRT-code), using
(165, 165+33, 2547191)-Net in Base 3 — Upper bound on s
There is no (165, 198, 2547192)-net in base 3, because
- 1 times m-reduction [i] would yield (165, 197, 2547192)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 9837 608459 420668 479533 850695 377911 346656 928346 416101 924119 371868 975175 367203 354639 619692 038145 > 3197 [i]