MinT - Best Known (159−77, 159, s)-Nets in Base 4

Best Known (159−77, 159, s)-Nets in Base 4

(159−77, 159, 104)-Net over F₄ — Constructive and digital

Digital (82, 159, 104)-net over F₄, using

t-expansion [i] based on digital (73, 159, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(159−77, 159, 130)-Net over F₄ — Digital

Digital (82, 159, 130)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−77, 159, 1564)-Net in Base 4 — Upper bound on s

There is no (82, 159, 1565)-net in base 4, because

1 times m-reduction [i] would yield (82, 158, 1565)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 133866 299827 171704 465920 963838 482421 049764 436308 920297 969250 125901 062175 576215 962326 641397 457600 > 4¹⁵⁸ [i]