MinT - Best Known (78, 78+115, s)-Nets in Base 4

Best Known (78, 78+115, s)-Nets in Base 4

(78, 78+115, 104)-Net over F₄ — Constructive and digital

Digital (78, 193, 104)-net over F₄, using

t-expansion [i] based on digital (73, 193, 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+115, 112)-Net over F₄ — Digital

Digital (78, 193, 112)-net over F₄, using

t-expansion [i] based on digital (73, 193, 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+115, 739)-Net in Base 4 — Upper bound on s

There is no (78, 193, 740)-net in base 4, because

1 times m-reduction [i] would yield (78, 192, 740)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42 261588 492086 640267 908813 453526 403906 376745 108721 253539 560479 496477 045402 282370 980910 437168 593919 726909 169644 553240 > 4¹⁹² [i]