Best Known (56−12, 56, s)-Nets in Base 3
(56−12, 56, 464)-Net over F3 — Constructive and digital
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
(56−12, 56, 1093)-Net over F3 — Digital
Digital (44, 56, 1093)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(356, 1093, F3, 2, 12) (dual of [(1093, 2), 2130, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(356, 2186, F3, 12) (dual of [2186, 2130, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(356, 2187, F3, 12) (dual of [2187, 2131, 13]-code), using
- 1 times truncation [i] based on linear OA(357, 2188, F3, 13) (dual of [2188, 2131, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(357, 2188, F3, 13) (dual of [2188, 2131, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(356, 2187, F3, 12) (dual of [2187, 2131, 13]-code), using
- OOA 2-folding [i] based on linear OA(356, 2186, F3, 12) (dual of [2186, 2130, 13]-code), using
(56−12, 56, 42488)-Net in Base 3 — Upper bound on s
There is no (44, 56, 42489)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 523 412151 297876 910043 069833 > 356 [i]