Best Known (189−30, 189, s)-Nets in Base 3
(189−30, 189, 1480)-Net over F3 — Constructive and digital
Digital (159, 189, 1480)-net over F3, using
- 31 times duplication [i] based on digital (158, 188, 1480)-net over F3, using
- t-expansion [i] based on digital (157, 188, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 47, 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, 47, 370)-net over F81, using
- t-expansion [i] based on digital (157, 188, 1480)-net over F3, using
(189−30, 189, 9862)-Net over F3 — Digital
Digital (159, 189, 9862)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3189, 9862, F3, 2, 30) (dual of [(9862, 2), 19535, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3189, 19724, F3, 30) (dual of [19724, 19535, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3189, 19725, F3, 30) (dual of [19725, 19536, 31]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- 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(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3189, 19725, F3, 30) (dual of [19725, 19536, 31]-code), using
- OOA 2-folding [i] based on linear OA(3189, 19724, F3, 30) (dual of [19724, 19535, 31]-code), using
(189−30, 189, 3299609)-Net in Base 3 — Upper bound on s
There is no (159, 189, 3299610)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 499402 111366 350671 838755 124210 874171 076164 918136 992046 193986 720475 724037 205440 201367 211961 > 3189 [i]