Best Known (215−44, 215, s)-Nets in Base 3
(215−44, 215, 688)-Net over F3 — Constructive and digital
Digital (171, 215, 688)-net over F3, using
- t-expansion [i] based on digital (169, 215, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (169, 216, 688)-net over F3, using
(215−44, 215, 2188)-Net over F3 — Digital
Digital (171, 215, 2188)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3215, 2188, F3, 44) (dual of [2188, 1973, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 2226, F3, 44) (dual of [2226, 2011, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(37) [i] based on
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(311, 39, F3, 5) (dual of [39, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(37, 14, F3, 5) (dual of [14, 7, 6]-code), using
- extended quadratic residue code Qe(14,3) [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- construction X applied to Ce(43) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3215, 2226, F3, 44) (dual of [2226, 2011, 45]-code), using
(215−44, 215, 208235)-Net in Base 3 — Upper bound on s
There is no (171, 215, 208236)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 811602 331994 723810 900940 453040 529924 621327 557281 699903 062634 861147 578740 356852 820536 229426 259520 862281 > 3215 [i]