MinT - Best Known (70, 70+38, s)-Nets in Base 25

Best Known (70, 70+38, s)-Nets in Base 25

(70, 70+38, 414)-Net over F₂₅ — Constructive and digital

Digital (70, 108, 414)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (10, 29, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ 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]
2. digital (41, 79, 288)-net over F₂₅, using
  - net from sequence [i] based on digital (41, 287)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 41 and N(F) ≥ 288, using
      - a function field by GarcÃa and Quoos [i]

(70, 70+38, 7813)-Net over F₂₅ — Digital

Digital (70, 108, 7813)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25¹⁰⁸, 7813, F₂₅, 38) (dual of [7813, 7705, 39]-code), using
- the BCH-code C(I) with length 7813Â | 25⁶âˆ’1, defining interval IÂ =Â {−25,821,1667,…,25}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]

(70, 70+38, large)-Net in Base 25 — Upper bound on s

There is no (70, 108, large)-net in base 25, because

36 times m-reduction [i] would yield (70, 72, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]