Best Known (77, 77+22, s)-Nets in Base 3
(77, 77+22, 464)-Net over F3 — Constructive and digital
Digital (77, 99, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (77, 100, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 25, 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, 25, 116)-net over F81, using
(77, 77+22, 1064)-Net over F3 — Digital
Digital (77, 99, 1064)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(399, 1064, F3, 2, 22) (dual of [(1064, 2), 2029, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(399, 1093, F3, 2, 22) (dual of [(1093, 2), 2087, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(399, 2186, F3, 22) (dual of [2186, 2087, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using
- OOA 2-folding [i] based on linear OA(399, 2186, F3, 22) (dual of [2186, 2087, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(399, 1093, F3, 2, 22) (dual of [(1093, 2), 2087, 23]-NRT-code), using
(77, 77+22, 48303)-Net in Base 3 — Upper bound on s
There is no (77, 99, 48304)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 171801 330763 984326 566415 483276 105576 090640 191681 > 399 [i]