Best Known (192, 192+46, s)-Nets in Base 3
(192, 192+46, 896)-Net over F3 — Constructive and digital
Digital (192, 238, 896)-net over F3, using
- 32 times duplication [i] based on digital (190, 236, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 59, 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, 59, 224)-net over F81, using
(192, 192+46, 3164)-Net over F3 — Digital
Digital (192, 238, 3164)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3238, 3164, F3, 46) (dual of [3164, 2926, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 3290, F3, 46) (dual of [3290, 3052, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(43) [i] based on
- linear OA(3237, 3281, F3, 46) (dual of [3281, 3044, 47]-code), using an extension Ce(45) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- 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(31, 9, F3, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(45) ⊂ Ce(43) [i] based on
- discarding factors / shortening the dual code based on linear OA(3238, 3290, F3, 46) (dual of [3290, 3052, 47]-code), using
(192, 192+46, 407915)-Net in Base 3 — Upper bound on s
There is no (192, 238, 407916)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 358817 351315 352266 071872 984867 389363 341389 667829 808870 113513 184314 958057 462984 150889 213726 382915 313582 382909 734481 > 3238 [i]