Best Known (72−14, 72, s)-Nets in Base 3
(72−14, 72, 600)-Net over F3 — Constructive and digital
Digital (58, 72, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 18, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
(72−14, 72, 1748)-Net over F3 — Digital
Digital (58, 72, 1748)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(372, 1748, F3, 14) (dual of [1748, 1676, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(372, 2217, F3, 14) (dual of [2217, 2145, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- linear OA(364, 2187, F3, 14) (dual of [2187, 2123, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(336, 2187, F3, 8) (dual of [2187, 2151, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-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, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(13) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(372, 2217, F3, 14) (dual of [2217, 2145, 15]-code), using
(72−14, 72, 136584)-Net in Base 3 — Upper bound on s
There is no (58, 72, 136585)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 22529 167844 931063 017505 600265 338795 > 372 [i]