Best Known (44−12, 44, s)-Nets in Base 3
(44−12, 44, 144)-Net over F3 — Constructive and digital
Digital (32, 44, 144)-net over F3, using
- 1 times m-reduction [i] based on digital (32, 45, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 15, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- trace code for nets [i] based on digital (2, 15, 48)-net over F27, using
(44−12, 44, 247)-Net over F3 — Digital
Digital (32, 44, 247)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(344, 247, F3, 12) (dual of [247, 203, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(344, 257, F3, 12) (dual of [257, 213, 13]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(341, 244, F3, 13) (dual of [244, 203, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(331, 244, F3, 9) (dual of [244, 213, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(344, 257, F3, 12) (dual of [257, 213, 13]-code), using
(44−12, 44, 4716)-Net in Base 3 — Upper bound on s
There is no (32, 44, 4717)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 986 001993 534276 432465 > 344 [i]