Best Known (161, 161+33, s)-Nets in Base 3
(161, 161+33, 896)-Net over F3 — Constructive and digital
Digital (161, 194, 896)-net over F3, using
- t-expansion [i] based on digital (160, 194, 896)-net over F3, using
- 2 times m-reduction [i] based on digital (160, 196, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 49, 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, 49, 224)-net over F81, using
- 2 times m-reduction [i] based on digital (160, 196, 896)-net over F3, using
(161, 161+33, 5771)-Net over F3 — Digital
Digital (161, 194, 5771)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3194, 5771, F3, 33) (dual of [5771, 5577, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3194, 6596, F3, 33) (dual of [6596, 6402, 34]-code), using
- construction X4 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(333, 34, F3, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,3)), using
- dual of repetition code with length 34 [i]
- linear OA(31, 34, F3, 1) (dual of [34, 33, 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 X4 applied to C([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3194, 6596, F3, 33) (dual of [6596, 6402, 34]-code), using
(161, 161+33, 1935443)-Net in Base 3 — Upper bound on s
There is no (161, 194, 1935444)-net in base 3, because
- 1 times m-reduction [i] would yield (161, 193, 1935444)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 121 452300 384489 696641 601578 854813 675081 370213 705504 583876 555045 784589 029414 760215 408987 616385 > 3193 [i]