Best Known (49−16, 49, s)-Nets in Base 5
(49−16, 49, 142)-Net over F5 — Constructive and digital
Digital (33, 49, 142)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (24, 40, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 20, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 20, 66)-net over F25, using
- digital (1, 9, 10)-net over F5, using
(49−16, 49, 368)-Net over F5 — Digital
Digital (33, 49, 368)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(549, 368, F5, 16) (dual of [368, 319, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(549, 624, F5, 16) (dual of [624, 575, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(549, 624, F5, 16) (dual of [624, 575, 17]-code), using
(49−16, 49, 17975)-Net in Base 5 — Upper bound on s
There is no (33, 49, 17976)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 17765 057756 405007 610983 222604 391425 > 549 [i]