Best Known (162, 162+43, s)-Nets in Base 3
(162, 162+43, 688)-Net over F3 — Constructive and digital
Digital (162, 205, 688)-net over F3, using
- 31 times duplication [i] based on digital (161, 204, 688)-net over F3, using
- t-expansion [i] based on digital (160, 204, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 51, 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, 51, 172)-net over F81, using
- t-expansion [i] based on digital (160, 204, 688)-net over F3, using
(162, 162+43, 1871)-Net over F3 — Digital
Digital (162, 205, 1871)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3205, 1871, F3, 43) (dual of [1871, 1666, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3205, 2216, F3, 43) (dual of [2216, 2011, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [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]
- 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(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3205, 2216, F3, 43) (dual of [2216, 2011, 44]-code), using
(162, 162+43, 187200)-Net in Base 3 — Upper bound on s
There is no (162, 205, 187201)-net in base 3, because
- 1 times m-reduction [i] would yield (162, 204, 187201)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21 515729 763464 722605 075099 146657 719585 588004 380878 854396 579474 096031 052279 718178 322673 388869 052723 > 3204 [i]