MinT - Best Known (56, 75, s)-Nets in Base 5

Best Known (56, 75, s)-Nets in Base 5

(56, 75, 280)-Net over F₅ — Constructive and digital

Digital (56, 75, 280)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (8, 17, 28)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 5, 12)-net over F₅, using
      - linear code embedding found in a computer search [i]
    2. digital (3, 12, 16)-net over F₅, using
      - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
2. digital (39, 58, 252)-net over F₅, using
  - trace code for nets [i] based on 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]

(56, 75, 1562)-Net over F₅ — Digital

Digital (56, 75, 1562)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷⁵, 1562, F₅, 19) (dual of [1562, 1487, 20]-code), using
- the BCH-code C(I) with length 1562Â | 5⁵âˆ’1, defining interval IÂ =Â {1,42,83,…,739}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(56, 75, 579126)-Net in Base 5 — Upper bound on s

There is no (56, 75, 579127)-net in base 5, because

1 times m-reduction [i] would yield (56, 74, 579127)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 5293 993315 402505 965210 758819 957939 738711 696208 296125 > 5⁷⁴ [i]