Best Known (207−41, 207, s)-Nets in Base 3
(207−41, 207, 688)-Net over F3 — Constructive and digital
Digital (166, 207, 688)-net over F3, using
- 5 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
(207−41, 207, 2389)-Net over F3 — Digital
Digital (166, 207, 2389)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3207, 2389, F3, 41) (dual of [2389, 2182, 42]-code), using
- 178 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 26 times 0, 1, 32 times 0, 1, 38 times 0) [i] based on linear OA(3190, 2194, F3, 41) (dual of [2194, 2004, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(39) [i] based on
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [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(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(40) ⊂ Ce(39) [i] based on
- 178 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 26 times 0, 1, 32 times 0, 1, 38 times 0) [i] based on linear OA(3190, 2194, F3, 41) (dual of [2194, 2004, 42]-code), using
(207−41, 207, 340879)-Net in Base 3 — Upper bound on s
There is no (166, 207, 340880)-net in base 3, because
- 1 times m-reduction [i] would yield (166, 206, 340880)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 193 643342 050760 149071 250356 143329 258020 860199 706286 559667 994533 814398 368333 847376 138461 979123 265921 > 3206 [i]