MinT - Best Known (81, 81+69, s)-Nets in Base 5

Best Known (81, 81+69, s)-Nets in Base 5

(81, 81+69, 132)-Net over F₅ — Constructive and digital

Digital (81, 150, 132)-net over F₅, using

t-expansion [i] based on digital (79, 150, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 75, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(81, 81+69, 194)-Net over F₅ — Digital

Digital (81, 150, 194)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+69, 3888)-Net in Base 5 — Upper bound on s

There is no (81, 150, 3889)-net in base 5, because

1 times m-reduction [i] would yield (81, 149, 3889)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 140 853726 663706 409570 944625 755819 002842 283700 320745 995731 208737 297097 977407 709635 762960 849531 976764 809625 > 5¹⁴⁹ [i]