MinT - Best Known (139−51, 139, s)-Nets in Base 5

Best Known (139−51, 139, s)-Nets in Base 5

(139−51, 139, 252)-Net over F₅ — Constructive and digital

Digital (88, 139, 252)-net over F₅, using

t-expansion [i] based on digital (85, 139, 252)-net over F₅, using
- 11 times m-reduction [i] based on digital (85, 150, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 75, 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]

(139−51, 139, 416)-Net over F₅ — Digital

Digital (88, 139, 416)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−51, 139, 18342)-Net in Base 5 — Upper bound on s

There is no (88, 139, 18343)-net in base 5, because

1 times m-reduction [i] would yield (88, 138, 18343)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 873467 872001 752432 675952 023601 001353 312915 869235 074729 408941 994512 858975 998956 355432 389745 122877 > 5¹³⁸ [i]