Best Known (68−21, 68, s)-Nets in Base 5
(68−21, 68, 252)-Net over F5 — Constructive and digital
Digital (47, 68, 252)-net over F5, using
- 6 times m-reduction [i] based on digital (47, 74, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 37, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 37, 126)-net over F25, using
(68−21, 68, 566)-Net over F5 — Digital
Digital (47, 68, 566)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(568, 566, F5, 21) (dual of [566, 498, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(568, 624, F5, 21) (dual of [624, 556, 22]-code), using
- the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(568, 624, F5, 21) (dual of [624, 556, 22]-code), using
(68−21, 68, 54570)-Net in Base 5 — Upper bound on s
There is no (47, 68, 54571)-net in base 5, because
- 1 times m-reduction [i] would yield (47, 67, 54571)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 67766 182928 869248 437595 980568 454728 838699 012441 > 567 [i]