MinT - Best Known (72, 72+69, s)-Nets in Base 5

Best Known (72, 72+69, s)-Nets in Base 5

(72, 72+69, 94)-Net over F₅ — Constructive and digital

Digital (72, 141, 94)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 41, 22)-net over F₅, using
  - net from sequence [i] based on digital (7, 21)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 7 and N(F) ≥ 22, 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]

(72, 72+69, 149)-Net over F₅ — Digital

Digital (72, 141, 149)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+69, 2530)-Net in Base 5 — Upper bound on s

There is no (72, 141, 2531)-net in base 5, because

1 times m-reduction [i] would yield (72, 140, 2531)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 71 852421 533140 834009 630205 916562 416292 700460 212870 337401 301053 823178 009476 927703 428241 440711 365625 > 5¹⁴⁰ [i]