MinT - Best Known (89−26, 89, s)-Nets in Base 5

Best Known (89−26, 89, s)-Nets in Base 5

(89−26, 89, 270)-Net over F₅ — Constructive and digital

Digital (63, 89, 270)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 17, 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 (46, 72, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 36, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(89−26, 89, 796)-Net over F₅ — Digital

Digital (63, 89, 796)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−26, 89, 86408)-Net in Base 5 — Upper bound on s

There is no (63, 89, 86409)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 161 573682 754575 009133 034886 942789 701763 851612 901745 845027 176885 > 5⁸⁹ [i]