MinT - Best Known (72, 72+75, s)-Nets in Base 5

Best Known (72, 72+75, s)-Nets in Base 5

(72, 72+75, 90)-Net over F₅ — Constructive and digital

Digital (72, 147, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 41, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 106, 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]

(72, 72+75, 133)-Net over F₅ — Digital

Digital (72, 147, 133)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+75, 2071)-Net in Base 5 — Upper bound on s

There is no (72, 147, 2072)-net in base 5, because

1 times m-reduction [i] would yield (72, 146, 2072)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 129525 976569 382998 187250 373420 169230 044990 289732 647983 543800 692249 570706 049565 937045 501694 823368 453985 > 5¹⁴⁶ [i]