MinT - Best Known (78, 78+181, s)-Nets in Base 4

Best Known (78, 78+181, s)-Nets in Base 4

(78, 78+181, 104)-Net over F₄ — Constructive and digital

Digital (78, 259, 104)-net over F₄, using

t-expansion [i] based on digital (73, 259, 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]

(78, 78+181, 112)-Net over F₄ — Digital

Digital (78, 259, 112)-net over F₄, using

t-expansion [i] based on digital (73, 259, 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]

(78, 78+181, 536)-Net in Base 4 — Upper bound on s

There is no (78, 259, 537)-net in base 4, because

1 times m-reduction [i] would yield (78, 258, 537)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 216289 135564 727063 415585 935717 588236 220933 305481 015954 720215 838089 592467 733711 451775 298559 685773 735102 005638 023126 149168 072389 965534 584335 132379 639889 147520 > 4²⁵⁸ [i]