MinT - Best Known (256−179, 256, s)-Nets in Base 4

Best Known (256−179, 256, s)-Nets in Base 4

(256−179, 256, 104)-Net over F₄ — Constructive and digital

Digital (77, 256, 104)-net over F₄, using

t-expansion [i] based on digital (73, 256, 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]

(256−179, 256, 112)-Net over F₄ — Digital

Digital (77, 256, 112)-net over F₄, using

t-expansion [i] based on digital (73, 256, 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]

(256−179, 256, 529)-Net in Base 4 — Upper bound on s

There is no (77, 256, 530)-net in base 4, because

1 times m-reduction [i] would yield (77, 255, 530)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3381 225660 754904 663007 950293 017836 713114 186427 645618 608910 926177 311789 188752 414473 842881 200778 355930 185953 227269 519117 611748 617591 469571 120727 371438 342848 > 4²⁵⁵ [i]