MinT - Best Known (72, 72+78, s)-Nets in Base 5

Best Known (72, 72+78, s)-Nets in Base 5

(72, 72+78, 84)-Net over F₅ — Constructive and digital

Digital (72, 150, 84)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 41, 12)-net over F₅, using
  - net from sequence [i] based on digital (2, 11)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 2 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 109, 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+78, 132)-Net over F₅ — Digital

Digital (72, 150, 132)-net over F₅, using

t-expansion [i] based on digital (67, 150, 132)-net over F₅, using
- net from sequence [i] based on digital (67, 131)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 67 and N(F) ≥ 132, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(72, 72+78, 1849)-Net in Base 5 — Upper bound on s

There is no (72, 150, 1850)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 708 338544 441424 436229 908568 221848 545999 901761 513846 687581 201711 203437 064203 641721 962581 569902 668693 022217 > 5¹⁵⁰ [i]