Best Known (45, 45+13, s)-Nets in Base 3
(45, 45+13, 400)-Net over F3 — Constructive and digital
Digital (45, 58, 400)-net over F3, using
- 32 times duplication [i] based on digital (43, 56, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 14, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 14, 100)-net over F81, using
(45, 45+13, 1055)-Net over F3 — Digital
Digital (45, 58, 1055)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(358, 1055, F3, 2, 13) (dual of [(1055, 2), 2052, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(358, 1097, F3, 2, 13) (dual of [(1097, 2), 2136, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(358, 2194, F3, 13) (dual of [2194, 2136, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(358, 2195, F3, 13) (dual of [2195, 2137, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [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(350, 2187, F3, 11) (dual of [2187, 2137, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(358, 2195, F3, 13) (dual of [2195, 2137, 14]-code), using
- OOA 2-folding [i] based on linear OA(358, 2194, F3, 13) (dual of [2194, 2136, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(358, 1097, F3, 2, 13) (dual of [(1097, 2), 2136, 14]-NRT-code), using
(45, 45+13, 51026)-Net in Base 3 — Upper bound on s
There is no (45, 58, 51027)-net in base 3, because
- 1 times m-reduction [i] would yield (45, 57, 51027)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1570 104398 673446 477642 490045 > 357 [i]