Best Known (60, 87, s)-Nets in Base 5
(60, 87, 258)-Net over F5 — Constructive and digital
Digital (60, 87, 258)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- 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
- digital (0, 13, 6)-net over F5, using
(60, 87, 629)-Net over F5 — Digital
Digital (60, 87, 629)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(587, 629, F5, 27) (dual of [629, 542, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 639, F5, 27) (dual of [639, 552, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(583, 625, F5, 27) (dual of [625, 542, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(573, 625, F5, 23) (dual of [625, 552, 24]-code), using an extension Ce(22) of 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]
- linear OA(54, 14, F5, 3) (dual of [14, 10, 4]-code or 14-cap in PG(3,5)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(587, 639, F5, 27) (dual of [639, 552, 28]-code), using
(60, 87, 59598)-Net in Base 5 — Upper bound on s
There is no (60, 87, 59599)-net in base 5, because
- 1 times m-reduction [i] would yield (60, 86, 59599)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1 292654 180420 640387 761402 648109 037548 826662 794449 824486 070029 > 586 [i]