Best Known (189, 189+36, s)-Nets in Base 3
(189, 189+36, 1480)-Net over F3 — Constructive and digital
Digital (189, 225, 1480)-net over F3, using
- t-expansion [i] based on digital (187, 225, 1480)-net over F3, using
- 3 times m-reduction [i] based on digital (187, 228, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 57, 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, 57, 370)-net over F81, using
- 3 times m-reduction [i] based on digital (187, 228, 1480)-net over F3, using
(189, 189+36, 9862)-Net over F3 — Digital
Digital (189, 225, 9862)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3225, 9862, F3, 2, 36) (dual of [(9862, 2), 19499, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3225, 19724, F3, 36) (dual of [19724, 19499, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 19725, F3, 36) (dual of [19725, 19500, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- linear OA(3217, 19684, F3, 37) (dual of [19684, 19467, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3181, 19684, F3, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3225, 19725, F3, 36) (dual of [19725, 19500, 37]-code), using
- OOA 2-folding [i] based on linear OA(3225, 19724, F3, 36) (dual of [19724, 19499, 37]-code), using
(189, 189+36, 3476270)-Net in Base 3 — Upper bound on s
There is no (189, 225, 3476271)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 225052 329489 697376 763739 551006 799245 852914 756432 551423 789316 372078 508170 520006 628101 345799 053897 684354 640957 > 3225 [i]