Best Known (37−12, 37, s)-Nets in Base 5
(37−12, 37, 132)-Net over F5 — Constructive and digital
Digital (25, 37, 132)-net over F5, using
- 5 times m-reduction [i] based on digital (25, 42, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 21, 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, 21, 66)-net over F25, using
(37−12, 37, 366)-Net over F5 — Digital
Digital (25, 37, 366)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(537, 366, F5, 12) (dual of [366, 329, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(537, 624, F5, 12) (dual of [624, 587, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(537, 624, F5, 12) (dual of [624, 587, 13]-code), using
(37−12, 37, 15288)-Net in Base 5 — Upper bound on s
There is no (25, 37, 15289)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 72 775217 863626 548366 309641 > 537 [i]