MinT - Best Known (68, 68+43, s)-Nets in Base 5

Best Known (68, 68+43, s)-Nets in Base 5

(68, 68+43, 252)-Net over F₅ — Constructive and digital

Digital (68, 111, 252)-net over F₅, using

5 times m-reduction [i] based on digital (68, 116, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 58, 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]

(68, 68+43, 280)-Net over F₅ — Digital

Digital (68, 111, 280)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+43, 9932)-Net in Base 5 — Upper bound on s

There is no (68, 111, 9933)-net in base 5, because

1 times m-reduction [i] would yield (68, 110, 9933)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 77194 912114 982890 168283 964376 716745 382897 785167 087363 576982 290480 900251 856165 > 5¹¹⁰ [i]