Best Known (71, 71+17, s)-Nets in Base 3
(71, 71+17, 640)-Net over F3 — Constructive and digital
Digital (71, 88, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 22, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
(71, 71+17, 1866)-Net over F3 — Digital
Digital (71, 88, 1866)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(388, 1866, F3, 17) (dual of [1866, 1778, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(388, 2200, F3, 17) (dual of [2200, 2112, 18]-code), using
- (u, u+v)-construction [i] based on
- linear OA(310, 13, F3, 8) (dual of [13, 3, 9]-code), using
- Simplex code S(3,3) [i]
- the expurgated narrow-sense BCH-code C(I) with length 13 | 33−1, defining interval I = [0,6], and minimum distance d ≥ |{−1,0,…,6}|+1 = 9 (BCH-bound) [i]
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(310, 13, F3, 8) (dual of [13, 3, 9]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(388, 2200, F3, 17) (dual of [2200, 2112, 18]-code), using
(71, 71+17, 290631)-Net in Base 3 — Upper bound on s
There is no (71, 88, 290632)-net in base 3, because
- 1 times m-reduction [i] would yield (71, 87, 290632)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 323263 404645 830724 628242 519345 024136 895745 > 387 [i]