Best Known (217−45, 217, s)-Nets in Base 3
(217−45, 217, 688)-Net over F3 — Constructive and digital
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
(217−45, 217, 2065)-Net over F3 — Digital
Digital (172, 217, 2065)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3217, 2065, F3, 45) (dual of [2065, 1848, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 2214, F3, 45) (dual of [2214, 1997, 46]-code), using
- construction X applied to Ce(45) ⊂ Ce(40) [i] based on
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [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(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(45) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3217, 2214, F3, 45) (dual of [2214, 1997, 46]-code), using
(217−45, 217, 218898)-Net in Base 3 — Upper bound on s
There is no (172, 217, 218899)-net in base 3, because
- 1 times m-reduction [i] would yield (172, 216, 218899)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 11 433909 852048 155753 276285 802679 771415 237938 344585 165884 592310 194651 823436 369796 319094 612774 481738 452637 > 3216 [i]