MinT - Best Known (143−69, 143, s)-Nets in Base 5

Best Known (143−69, 143, s)-Nets in Base 5

(143−69, 143, 98)-Net over F₅ — Constructive and digital

Digital (74, 143, 98)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (9, 43, 26)-net over F₅, using
  - net from sequence [i] based on digital (9, 25)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 9 and N(F) ≥ 26, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 100, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(143−69, 143, 158)-Net over F₅ — Digital

Digital (74, 143, 158)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(143−69, 143, 2784)-Net in Base 5 — Upper bound on s

There is no (74, 143, 2785)-net in base 5, because

1 times m-reduction [i] would yield (74, 142, 2785)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1800 835217 221795 008278 711924 351128 434479 294579 409407 393351 287632 987375 823061 571843 912652 968442 199961 > 5¹⁴² [i]