MinT - Best Known (127−41, 127, s)-Nets in Base 4

Best Known (127−41, 127, s)-Nets in Base 4

(127−41, 127, 240)-Net over F₄ — Constructive and digital

Digital (86, 127, 240)-net over F₄, using

4¹ times duplication [i] based on digital (85, 126, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 42, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(127−41, 127, 426)-Net over F₄ — Digital

Digital (86, 127, 426)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−41, 127, 17169)-Net in Base 4 — Upper bound on s

There is no (86, 127, 17170)-net in base 4, because

1 times m-reduction [i] would yield (86, 126, 17170)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7242 322994 339241 333642 099581 893356 320555 481569 813016 559054 163677 537091 662144 > 4¹²⁶ [i]