MinT - Best Known (80, 80+107, s)-Nets in Base 4

Best Known (80, 80+107, s)-Nets in Base 4

(80, 80+107, 104)-Net over F₄ — Constructive and digital

Digital (80, 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]

(80, 80+107, 112)-Net over F₄ — Digital

Digital (80, 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]

(80, 80+107, 847)-Net in Base 4 — Upper bound on s

There is no (80, 187, 848)-net in base 4, because

1 times m-reduction [i] would yield (80, 186, 848)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9922 696723 545748 022443 900719 235562 816519 409154 193743 357550 501429 383752 421297 932850 695514 142411 191711 029343 318900 > 4¹⁸⁶ [i]