Best Known (193−41, 193, s)-Nets in Base 3
(193−41, 193, 688)-Net over F3 — Constructive and digital
Digital (152, 193, 688)-net over F3, using
- 31 times duplication [i] based on digital (151, 192, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 172)-net over F81, using
(193−41, 193, 1683)-Net over F3 — Digital
Digital (152, 193, 1683)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3193, 1683, F3, 41) (dual of [1683, 1490, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3193, 2200, F3, 41) (dual of [2200, 2007, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(37) [i] based on
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [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(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(40) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3193, 2200, F3, 41) (dual of [2200, 2007, 42]-code), using
(193−41, 193, 157974)-Net in Base 3 — Upper bound on s
There is no (152, 193, 157975)-net in base 3, because
- 1 times m-reduction [i] would yield (152, 192, 157975)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 40 486908 665464 309812 549532 037902 441090 325240 245929 269238 801896 959953 130531 644027 886281 532761 > 3192 [i]