Best Known (224−41, 224, s)-Nets in Base 3
(224−41, 224, 896)-Net over F3 — Constructive and digital
Digital (183, 224, 896)-net over F3, using
- t-expansion [i] based on digital (181, 224, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 56, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 56, 224)-net over F81, using
(224−41, 224, 4080)-Net over F3 — Digital
Digital (183, 224, 4080)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3224, 4080, F3, 41) (dual of [4080, 3856, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 6592, F3, 41) (dual of [6592, 6368, 42]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3223, 6591, F3, 41) (dual of [6591, 6368, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-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
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3223, 6591, F3, 41) (dual of [6591, 6368, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 6592, F3, 41) (dual of [6592, 6368, 42]-code), using
(224−41, 224, 867298)-Net in Base 3 — Upper bound on s
There is no (183, 224, 867299)-net in base 3, because
- 1 times m-reduction [i] would yield (183, 223, 867299)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 25006 188195 632888 943442 866763 715610 809617 672595 579413 597895 540914 537016 391127 011893 277918 272398 459179 629241 > 3223 [i]