Best Known (51, 51+12, s)-Nets in Base 3
(51, 51+12, 471)-Net over F3 — Constructive and digital
Digital (51, 63, 471)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- a shift-net [i]
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (44, 56, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 14, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 14, 116)-net over F81, using
- digital (1, 7, 7)-net over F3, using
(51, 51+12, 2047)-Net over F3 — Digital
Digital (51, 63, 2047)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(363, 2047, F3, 12) (dual of [2047, 1984, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(363, 2214, F3, 12) (dual of [2214, 2151, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(357, 2187, F3, 13) (dual of [2187, 2130, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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, 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(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(363, 2214, F3, 12) (dual of [2214, 2151, 13]-code), using
(51, 51+12, 153090)-Net in Base 3 — Upper bound on s
There is no (51, 63, 153091)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 144561 279223 255645 154300 569757 > 363 [i]