Best Known (230−44, 230, s)-Nets in Base 3
(230−44, 230, 896)-Net over F3 — Constructive and digital
Digital (186, 230, 896)-net over F3, using
- 32 times duplication [i] based on digital (184, 228, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 57, 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, 57, 224)-net over F81, using
(230−44, 230, 3258)-Net over F3 — Digital
Digital (186, 230, 3258)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3230, 3258, F3, 44) (dual of [3258, 3028, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3230, 3290, F3, 44) (dual of [3290, 3060, 45]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3229, 3289, F3, 44) (dual of [3289, 3060, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3229, 3281, F3, 44) (dual of [3281, 3052, 45]-code), using an extension Ce(43) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3221, 3281, F3, 43) (dual of [3281, 3060, 44]-code), using an extension Ce(42) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3229, 3289, F3, 44) (dual of [3289, 3060, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3230, 3290, F3, 44) (dual of [3290, 3060, 45]-code), using
(230−44, 230, 440440)-Net in Base 3 — Upper bound on s
There is no (186, 230, 440441)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 54 687739 285941 545361 834632 053193 398586 346731 808801 982802 261427 588923 472225 481512 342861 041432 884872 195895 282377 > 3230 [i]