MinT - Best Known (59, 59+22, s)-Nets in Base 5

Best Known (59, 59+22, s)-Nets in Base 5

(59, 59+22, 273)-Net over F₅ — Constructive and digital

Digital (59, 81, 273)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (6, 17, 21)-net over F₅, using
  - net from sequence [i] based on digital (6, 20)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 6 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (42, 64, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 32, 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]

(59, 59+22, 1088)-Net over F₅ — Digital

Digital (59, 81, 1088)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+22, 172145)-Net in Base 5 — Upper bound on s

There is no (59, 81, 172146)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 413 604774 532611 482018 388280 502135 392880 875298 033334 557545 > 5⁸¹ [i]