Best Known (50, 72, s)-Nets in Base 5
(50, 72, 252)-Net over F5 — Constructive and digital
Digital (50, 72, 252)-net over F5, using
- 8 times m-reduction [i] based on digital (50, 80, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 40, 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, 40, 126)-net over F25, using
(50, 72, 616)-Net over F5 — Digital
Digital (50, 72, 616)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(572, 616, F5, 22) (dual of [616, 544, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 624, F5, 22) (dual of [624, 552, 23]-code), using
- the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(572, 624, F5, 22) (dual of [624, 552, 23]-code), using
(50, 72, 46127)-Net in Base 5 — Upper bound on s
There is no (50, 72, 46128)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 211 795911 582198 166763 224403 439875 234825 221849 452225 > 572 [i]