Best Known (168, 168+36, s)-Nets in Base 3
(168, 168+36, 896)-Net over F3 — Constructive and digital
Digital (168, 204, 896)-net over F3, using
- t-expansion [i] based on digital (166, 204, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 51, 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, 51, 224)-net over F81, using
(168, 168+36, 4744)-Net over F3 — Digital
Digital (168, 204, 4744)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3204, 4744, F3, 36) (dual of [4744, 4540, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3204, 6605, F3, 36) (dual of [6605, 6401, 37]-code), using
- strength reduction [i] based on linear OA(3204, 6605, F3, 37) (dual of [6605, 6401, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [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(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(311, 43, F3, 5) (dual of [43, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- strength reduction [i] based on linear OA(3204, 6605, F3, 37) (dual of [6605, 6401, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3204, 6605, F3, 36) (dual of [6605, 6401, 37]-code), using
(168, 168+36, 964864)-Net in Base 3 — Upper bound on s
There is no (168, 204, 964865)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 21 514865 495954 597770 795553 757549 582546 968860 849173 025541 399835 297474 900110 864968 368259 637463 706441 > 3204 [i]