Best Known (162, 162+42, s)-Nets in Base 3
(162, 162+42, 688)-Net over F3 — Constructive and digital
Digital (162, 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
(162, 162+42, 2043)-Net over F3 — Digital
Digital (162, 204, 2043)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3204, 2043, F3, 42) (dual of [2043, 1839, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3204, 2216, F3, 42) (dual of [2216, 2012, 43]-code), using
- construction XX applied to Ce(42) ⊂ Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(42) ⊂ Ce(37) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(3204, 2216, F3, 42) (dual of [2216, 2012, 43]-code), using
(162, 162+42, 187200)-Net in Base 3 — Upper bound on s
There is no (162, 204, 187201)-net in base 3, because
- 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]