MinT - Best Known (68, 68+45, s)-Nets in Base 5

Best Known (68, 68+45, s)-Nets in Base 5

(68, 68+45, 252)-Net over F₅ — Constructive and digital

Digital (68, 113, 252)-net over F₅, using

3 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+45, 256)-Net over F₅ — Digital

Digital (68, 113, 256)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+45, 8172)-Net in Base 5 — Upper bound on s

There is no (68, 113, 8173)-net in base 5, because

1 times m-reduction [i] would yield (68, 112, 8173)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 930275 246364 276315 038212 838408 747155 437966 034049 368468 073937 043696 541332 283721 > 5¹¹² [i]