Best Known (171, 212, s)-Nets in Base 3
(171, 212, 692)-Net over F3 — Constructive and digital
Digital (171, 212, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 20, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (151, 192, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 172)-net over F81, using
- digital (0, 20, 4)-net over F3, using
(171, 212, 2687)-Net over F3 — Digital
Digital (171, 212, 2687)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3212, 2687, F3, 41) (dual of [2687, 2475, 42]-code), using
- 471 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, 1, 46 times 0, 1, 52 times 0, 1, 59 times 0, 1, 63 times 0, 1, 68 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
- 471 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, 1, 46 times 0, 1, 52 times 0, 1, 59 times 0, 1, 63 times 0, 1, 68 times 0) [i] based on linear OA(3190, 2194, F3, 41) (dual of [2194, 2004, 42]-code), using
(171, 212, 448628)-Net in Base 3 — Upper bound on s
There is no (171, 212, 448629)-net in base 3, because
- 1 times m-reduction [i] would yield (171, 211, 448629)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 47054 340531 500507 104069 243491 017562 519040 525428 526791 124037 868845 166963 783918 490656 350690 559331 973521 > 3211 [i]