Best Known (170−35, 170, s)-Nets in Base 3
(170−35, 170, 688)-Net over F3 — Constructive and digital
Digital (135, 170, 688)-net over F3, using
- 32 times duplication [i] based on digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
(170−35, 170, 1796)-Net over F3 — Digital
Digital (135, 170, 1796)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3170, 1796, F3, 35) (dual of [1796, 1626, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3170, 2203, F3, 35) (dual of [2203, 2033, 36]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3155, 2188, F3, 33) (dual of [2188, 2033, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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(3170, 2203, F3, 35) (dual of [2203, 2033, 36]-code), using
(170−35, 170, 198621)-Net in Base 3 — Upper bound on s
There is no (135, 170, 198622)-net in base 3, because
- 1 times m-reduction [i] would yield (135, 169, 198622)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 430 050690 406718 544759 574045 551642 831076 392513 633271 623549 610573 063170 707543 370141 > 3169 [i]