Best Known (48, 48+13, s)-Nets in Base 3
(48, 48+13, 464)-Net over F3 — Constructive and digital
Digital (48, 61, 464)-net over F3, using
- 31 times duplication [i] based on digital (47, 60, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 15, 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, 15, 116)-net over F81, using
(48, 48+13, 1102)-Net over F3 — Digital
Digital (48, 61, 1102)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(361, 1102, F3, 2, 13) (dual of [(1102, 2), 2143, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(361, 2204, F3, 13) (dual of [2204, 2143, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(361, 2205, F3, 13) (dual of [2205, 2144, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [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(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(34, 18, F3, 2) (dual of [18, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(361, 2205, F3, 13) (dual of [2205, 2144, 14]-code), using
- OOA 2-folding [i] based on linear OA(361, 2204, F3, 13) (dual of [2204, 2143, 14]-code), using
(48, 48+13, 88384)-Net in Base 3 — Upper bound on s
There is no (48, 61, 88385)-net in base 3, because
- 1 times m-reduction [i] would yield (48, 60, 88385)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 42391 717936 676859 410873 864025 > 360 [i]