Best Known (250−53, 250, s)-Nets in Base 3
(250−53, 250, 688)-Net over F3 — Constructive and digital
Digital (197, 250, 688)-net over F3, using
- 32 times duplication [i] based on digital (195, 248, 688)-net over F3, using
- t-expansion [i] based on digital (193, 248, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 62, 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, 62, 172)-net over F81, using
- t-expansion [i] based on digital (193, 248, 688)-net over F3, using
(250−53, 250, 2073)-Net over F3 — Digital
Digital (197, 250, 2073)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3250, 2073, F3, 53) (dual of [2073, 1823, 54]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 2205, F3, 53) (dual of [2205, 1955, 54]-code), using
- construction X applied to Ce(52) ⊂ Ce(49) [i] based on
- linear OA(3246, 2187, F3, 53) (dual of [2187, 1941, 54]-code), using an extension Ce(52) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,52], and designed minimum distance d ≥ |I|+1 = 53 [i]
- linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(34, 18, F3, 2) (dual of [18, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(52) ⊂ Ce(49) [i] based on
- discarding factors / shortening the dual code based on linear OA(3250, 2205, F3, 53) (dual of [2205, 1955, 54]-code), using
(250−53, 250, 195686)-Net in Base 3 — Upper bound on s
There is no (197, 250, 195687)-net in base 3, because
- 1 times m-reduction [i] would yield (197, 249, 195687)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 63568 718943 040761 461344 931517 083483 907779 491400 439445 444376 050294 761907 352949 453014 420518 604904 520414 115103 682878 685725 > 3249 [i]