Best Known (41, 41+11, s)-Nets in Base 3
(41, 41+11, 464)-Net over F3 — Constructive and digital
Digital (41, 52, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 13, 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
(41, 41+11, 1098)-Net over F3 — Digital
Digital (41, 52, 1098)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(352, 1098, F3, 2, 11) (dual of [(1098, 2), 2144, 12]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(350, 1097, F3, 2, 11) (dual of [(1097, 2), 2144, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(350, 2194, F3, 11) (dual of [2194, 2144, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(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(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(350, 2194, F3, 11) (dual of [2194, 2144, 12]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(350, 1097, F3, 2, 11) (dual of [(1097, 2), 2144, 12]-NRT-code), using
(41, 41+11, 95812)-Net in Base 3 — Upper bound on s
There is no (41, 52, 95813)-net in base 3, because
- 1 times m-reduction [i] would yield (41, 51, 95813)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 153738 180732 061480 209307 > 351 [i]