MinT - Best Known (138−61, 138, s)-Nets in Base 5

Best Known (138−61, 138, s)-Nets in Base 5

(138−61, 138, 132)-Net over F₅ — Constructive and digital

Digital (77, 138, 132)-net over F₅, using

8 times m-reduction [i] based on digital (77, 146, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 73, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(138−61, 138, 207)-Net over F₅ — Digital

Digital (77, 138, 207)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−61, 138, 4662)-Net in Base 5 — Upper bound on s

There is no (77, 138, 4663)-net in base 5, because

1 times m-reduction [i] would yield (77, 137, 4663)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 574546 038489 826464 123140 325766 088643 821682 756066 405086 636150 199369 621942 181221 342743 832457 953417 > 5¹³⁷ [i]