Best Known (169, 212, s)-Nets in Base 3
(169, 212, 688)-Net over F3 — Constructive and digital
Digital (169, 212, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 54, 172)-net over F81, using
(169, 212, 2265)-Net over F3 — Digital
Digital (169, 212, 2265)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3212, 2265, F3, 43) (dual of [2265, 2053, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3212, 2273, F3, 43) (dual of [2273, 2061, 44]-code), using
- 70 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0) [i] based on linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- 70 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0) [i] based on linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3212, 2273, F3, 43) (dual of [2273, 2061, 44]-code), using
(169, 212, 269999)-Net in Base 3 — Upper bound on s
There is no (169, 212, 270000)-net in base 3, because
- 1 times m-reduction [i] would yield (169, 211, 270000)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 47056 265204 902044 138863 450884 744520 759611 717966 766318 345423 601703 198437 033307 226269 769899 410781 212001 > 3211 [i]