Best Known (213−44, 213, s)-Nets in Base 3
(213−44, 213, 688)-Net over F3 — Constructive and digital
Digital (169, 213, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
(213−44, 213, 2075)-Net over F3 — Digital
Digital (169, 213, 2075)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3213, 2075, F3, 44) (dual of [2075, 1862, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3213, 2219, F3, 44) (dual of [2219, 2006, 45]-code), using
- construction XX applied to Ce(43) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [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(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(36, 29, F3, 3) (dual of [29, 23, 4]-code or 29-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
- linear OA(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(43) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3213, 2219, F3, 44) (dual of [2219, 2006, 45]-code), using
(213−44, 213, 188440)-Net in Base 3 — Upper bound on s
There is no (169, 213, 188441)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 423488 979034 225221 426561 825046 038825 157316 299530 010110 959887 950793 694096 048448 777963 161232 838623 744777 > 3213 [i]