MinT - Best Known (257−179, 257, s)-Nets in Base 4

Best Known (257−179, 257, s)-Nets in Base 4

(257−179, 257, 104)-Net over F₄ — Constructive and digital

Digital (78, 257, 104)-net over F₄, using

t-expansion [i] based on digital (73, 257, 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]

(257−179, 257, 112)-Net over F₄ — Digital

Digital (78, 257, 112)-net over F₄, using

t-expansion [i] based on digital (73, 257, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(257−179, 257, 539)-Net in Base 4 — Upper bound on s

There is no (78, 257, 540)-net in base 4, because

1 times m-reduction [i] would yield (78, 256, 540)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14814 996197 424682 945042 803857 921291 272652 734125 164781 591379 057494 718333 219034 203707 744882 402240 102965 663872 248759 741047 733539 481705 250854 148183 553395 075864 > 4²⁵⁶ [i]