MinT - Best Known (78, 78+109, s)-Nets in Base 4

Best Known (78, 78+109, s)-Nets in Base 4

(78, 78+109, 104)-Net over F₄ — Constructive and digital

Digital (78, 187, 104)-net over F₄, using

t-expansion [i] based on digital (73, 187, 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+109, 112)-Net over F₄ — Digital

Digital (78, 187, 112)-net over F₄, using

t-expansion [i] based on digital (73, 187, 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+109, 784)-Net in Base 4 — Upper bound on s

There is no (78, 187, 785)-net in base 4, because

1 times m-reduction [i] would yield (78, 186, 785)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9821 468563 461789 477108 888830 892121 159019 632008 011730 119930 962445 590438 506763 315619 044914 806354 138468 017878 620800 > 4¹⁸⁶ [i]