Best Known (159, 159+35, s)-Nets in Base 3
(159, 159+35, 896)-Net over F3 — Constructive and digital
Digital (159, 194, 896)-net over F3, using
- 32 times duplication [i] based on digital (157, 192, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 48, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 48, 224)-net over F81, using
(159, 159+35, 4031)-Net over F3 — Digital
Digital (159, 194, 4031)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3194, 4031, F3, 35) (dual of [4031, 3837, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3194, 6579, F3, 35) (dual of [6579, 6385, 36]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3194, 6579, F3, 35) (dual of [6579, 6385, 36]-code), using
(159, 159+35, 936778)-Net in Base 3 — Upper bound on s
There is no (159, 194, 936779)-net in base 3, because
- 1 times m-reduction [i] would yield (159, 193, 936779)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 121 451516 022188 582462 053153 645826 554567 483862 680739 776634 648161 057751 018777 652259 130963 367159 > 3193 [i]