MinT - Best Known (85−22, 85, s)-Nets in Base 5

Best Known (85−22, 85, s)-Nets in Base 5

(85−22, 85, 296)-Net over F₅ — Constructive and digital

Digital (63, 85, 296)-net over F₅, using

3 times m-reduction [i] based on digital (63, 88, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 44, 148)-net over F₂₅, using
  - net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
      - a function field by GarcÃa and Quoos [i]

(85−22, 85, 1562)-Net over F₅ — Digital

Digital (63, 85, 1562)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁵, 1562, F₅, 22) (dual of [1562, 1477, 23]-code), using
- the BCH-code C(I) with length 1562Â | 5⁵âˆ’1, defining interval IÂ =Â {125,266,407,…,−38}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(85−22, 85, 309083)-Net in Base 5 — Upper bound on s

There is no (63, 85, 309084)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 258503 086606 482807 900632 859284 994171 950803 682030 440739 134065 > 5⁸⁵ [i]