MinT - Best Known (81, 81+39, s)-Nets in Base 4

Best Known (81, 81+39, s)-Nets in Base 4

(81, 81+39, 240)-Net over F₄ — Constructive and digital

Digital (81, 120, 240)-net over F₄, using

trace code for nets [i] based on digital (1, 40, 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]

(81, 81+39, 397)-Net over F₄ — Digital

Digital (81, 120, 397)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+39, 15576)-Net in Base 4 — Upper bound on s

There is no (81, 120, 15577)-net in base 4, because

1 times m-reduction [i] would yield (81, 119, 15577)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 442096 658356 570001 705655 863906 296472 699948 000838 143676 705890 652816 842536 > 4¹¹⁹ [i]