Best Known (238−51, 238, s)-Nets in Base 3
(238−51, 238, 688)-Net over F3 — Constructive and digital
Digital (187, 238, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (187, 240, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 60, 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, 60, 172)-net over F81, using
(238−51, 238, 1896)-Net over F3 — Digital
Digital (187, 238, 1896)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3238, 1896, F3, 51) (dual of [1896, 1658, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 2186, F3, 51) (dual of [2186, 1948, 52]-code), using
- 1 times truncation [i] based on linear OA(3239, 2187, F3, 52) (dual of [2187, 1948, 53]-code), using
- an extension Ce(51) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,51], and designed minimum distance d ≥ |I|+1 = 52 [i]
- 1 times truncation [i] based on linear OA(3239, 2187, F3, 52) (dual of [2187, 1948, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 2186, F3, 51) (dual of [2186, 1948, 52]-code), using
(238−51, 238, 169684)-Net in Base 3 — Upper bound on s
There is no (187, 238, 169685)-net in base 3, because
- 1 times m-reduction [i] would yield (187, 237, 169685)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 119607 377068 649756 909578 297664 789198 626021 712100 490808 860212 674548 489363 334764 246949 105892 362543 249911 878223 247307 > 3237 [i]