MinT - Best Known (67−19, 67, s)-Nets in Base 4

Best Known (67−19, 67, s)-Nets in Base 4

(67−19, 67, 312)-Net over F₄ — Constructive and digital

Digital (48, 67, 312)-net over F₄, using

4¹ times duplication [i] based on digital (47, 66, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 22, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(67−19, 67, 447)-Net over F₄ — Digital

Digital (48, 67, 447)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−19, 67, 35946)-Net in Base 4 — Upper bound on s

There is no (48, 67, 35947)-net in base 4, because

1 times m-reduction [i] would yield (48, 66, 35947)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5445 692667 901486 312344 861602 934830 889754 > 4⁶⁶ [i]