Best Known (37−20, 37, s)-Nets in Base 25
(37−20, 37, 126)-Net over F25 — Constructive and digital
Digital (17, 37, 126)-net over F25, using
- t-expansion [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
(37−20, 37, 189)-Net over F25 — Digital
Digital (17, 37, 189)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2537, 189, F25, 20) (dual of [189, 152, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2537, 208, F25, 20) (dual of [208, 171, 21]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 208 | 252−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2537, 208, F25, 20) (dual of [208, 171, 21]-code), using
(37−20, 37, 28058)-Net in Base 25 — Upper bound on s
There is no (17, 37, 28059)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 5294 588007 444508 978607 053627 340311 731141 202840 637905 > 2537 [i]