Best Known (28, 28+13, s)-Nets in Base 5
(28, 28+13, 142)-Net over F5 — Constructive and digital
Digital (28, 41, 142)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (21, 34, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 17, 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, 17, 66)-net over F25, using
- digital (1, 7, 10)-net over F5, using
(28, 28+13, 420)-Net over F5 — Digital
Digital (28, 41, 420)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(541, 420, F5, 13) (dual of [420, 379, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(541, 624, F5, 13) (dual of [624, 583, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(541, 624, F5, 13) (dual of [624, 583, 14]-code), using
(28, 28+13, 34190)-Net in Base 5 — Upper bound on s
There is no (28, 41, 34191)-net in base 5, because
- 1 times m-reduction [i] would yield (28, 40, 34191)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 9095 008017 778034 200808 191465 > 540 [i]