Best Known (217−43, 217, s)-Nets in Base 3
(217−43, 217, 688)-Net over F3 — Constructive and digital
Digital (174, 217, 688)-net over F3, using
- t-expansion [i] based on digital (172, 217, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (172, 220, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 55, 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, 55, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (172, 220, 688)-net over F3, using
(217−43, 217, 2453)-Net over F3 — Digital
Digital (174, 217, 2453)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3217, 2453, F3, 43) (dual of [2453, 2236, 44]-code), using
- 239 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0, 1, 12 times 0, 1, 17 times 0, 1, 21 times 0, 1, 27 times 0, 1, 33 times 0, 1, 40 times 0, 1, 46 times 0) [i] based on linear OA(3198, 2195, F3, 43) (dual of [2195, 1997, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(40) [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(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(31, 8, F3, 1) (dual of [8, 7, 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(40) [i] based on
- 239 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0, 1, 12 times 0, 1, 17 times 0, 1, 21 times 0, 1, 27 times 0, 1, 33 times 0, 1, 40 times 0, 1, 46 times 0) [i] based on linear OA(3198, 2195, F3, 43) (dual of [2195, 1997, 44]-code), using
(217−43, 217, 350727)-Net in Base 3 — Upper bound on s
There is no (174, 217, 350728)-net in base 3, because
- 1 times m-reduction [i] would yield (174, 216, 350728)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 11 434001 767075 818435 693544 665695 927799 901959 751833 797234 909587 765105 943529 032176 721146 441319 995819 379857 > 3216 [i]