Best Known (59, 59+28, s)-Nets in Base 5
(59, 59+28, 252)-Net over F5 — Constructive and digital
Digital (59, 87, 252)-net over F5, using
- 11 times m-reduction [i] based on digital (59, 98, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 49, 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, 49, 126)-net over F25, using
(59, 59+28, 524)-Net over F5 — Digital
Digital (59, 87, 524)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(587, 524, F5, 28) (dual of [524, 437, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 624, F5, 28) (dual of [624, 537, 29]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(587, 624, F5, 28) (dual of [624, 537, 29]-code), using
(59, 59+28, 33332)-Net in Base 5 — Upper bound on s
There is no (59, 87, 33333)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 6 463931 426151 546583 816592 691073 148922 772918 616858 508000 856425 > 587 [i]