Best Known (209−42, 209, s)-Nets in Base 3
(209−42, 209, 688)-Net over F3 — Constructive and digital
Digital (167, 209, 688)-net over F3, using
- t-expansion [i] based on digital (166, 209, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (166, 212, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 53, 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, 53, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (166, 212, 688)-net over F3, using
(209−42, 209, 2288)-Net over F3 — Digital
Digital (167, 209, 2288)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3209, 2288, F3, 42) (dual of [2288, 2079, 43]-code), using
- 75 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 16 times 0, 1, 23 times 0) [i] based on linear OA(3198, 2202, F3, 42) (dual of [2202, 2004, 43]-code), using
- construction X applied to Ce(42) ⊂ Ce(39) [i] based on
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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(42) ⊂ Ce(39) [i] based on
- 75 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 16 times 0, 1, 23 times 0) [i] based on linear OA(3198, 2202, F3, 42) (dual of [2202, 2004, 43]-code), using
(209−42, 209, 243174)-Net in Base 3 — Upper bound on s
There is no (167, 209, 243175)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5228 200061 420619 902187 196836 104921 476958 599559 453058 159164 347777 973321 042010 180717 653382 232972 224231 > 3209 [i]