MinT - Best Known (84, 84+87, s)-Nets in Base 4

Best Known (84, 84+87, s)-Nets in Base 4

(84, 84+87, 104)-Net over F₄ — Constructive and digital

Digital (84, 171, 104)-net over F₄, using

t-expansion [i] based on digital (73, 171, 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]

(84, 84+87, 129)-Net over F₄ — Digital

Digital (84, 171, 129)-net over F₄, using

t-expansion [i] based on digital (81, 171, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(84, 84+87, 1315)-Net in Base 4 — Upper bound on s

There is no (84, 171, 1316)-net in base 4, because

1 times m-reduction [i] would yield (84, 170, 1316)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 254386 104949 463544 402789 896981 828593 413198 522552 087587 797361 947104 141125 246619 617522 091912 768745 058720 > 4¹⁷⁰ [i]