Best Known (55, 55+25, s)-Nets in Base 5
(55, 55+25, 252)-Net over F5 — Constructive and digital
Digital (55, 80, 252)-net over F5, using
- 10 times m-reduction [i] based on digital (55, 90, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 45, 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, 45, 126)-net over F25, using
(55, 55+25, 578)-Net over F5 — Digital
Digital (55, 80, 578)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(580, 578, F5, 25) (dual of [578, 498, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(580, 624, F5, 25) (dual of [624, 544, 26]-code), using
- the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(580, 624, F5, 25) (dual of [624, 544, 26]-code), using
(55, 55+25, 52818)-Net in Base 5 — Upper bound on s
There is no (55, 80, 52819)-net in base 5, because
- 1 times m-reduction [i] would yield (55, 79, 52819)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 16 546402 574460 052090 852217 891519 912718 655434 036503 357009 > 579 [i]