Best Known (69−22, 69, s)-Nets in Base 5
(69−22, 69, 252)-Net over F5 — Constructive and digital
Digital (47, 69, 252)-net over F5, using
- 5 times m-reduction [i] based on 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
(69−22, 69, 481)-Net over F5 — Digital
Digital (47, 69, 481)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(569, 481, F5, 22) (dual of [481, 412, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(569, 624, F5, 22) (dual of [624, 555, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(569, 624, F5, 22) (dual of [624, 555, 23]-code), using
(69−22, 69, 29736)-Net in Base 5 — Upper bound on s
There is no (47, 69, 29737)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1 694406 547146 844672 231967 344112 056956 735986 802429 > 569 [i]