Best Known (44, 44+20, s)-Nets in Base 5
(44, 44+20, 252)-Net over F5 — Constructive and digital
Digital (44, 64, 252)-net over F5, using
- 4 times m-reduction [i] based on digital (44, 68, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 34, 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, 34, 126)-net over F25, using
(44, 44+20, 516)-Net over F5 — Digital
Digital (44, 64, 516)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(564, 516, F5, 20) (dual of [516, 452, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(564, 624, F5, 20) (dual of [624, 560, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(564, 624, F5, 20) (dual of [624, 560, 21]-code), using
(44, 44+20, 33669)-Net in Base 5 — Upper bound on s
There is no (44, 64, 33670)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 542 214066 210046 335216 875361 491061 611286 884353 > 564 [i]