MinT - Best Known (84, 84+144, s)-Nets in Base 4

Best Known (84, 84+144, s)-Nets in Base 4

(84, 84+144, 104)-Net over F₄ — Constructive and digital

Digital (84, 228, 104)-net over F₄, using

t-expansion [i] based on digital (73, 228, 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+144, 129)-Net over F₄ — Digital

Digital (84, 228, 129)-net over F₄, using

t-expansion [i] based on digital (81, 228, 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+144, 685)-Net in Base 4 — Upper bound on s

There is no (84, 228, 686)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 203585 843304 578858 201257 550373 484999 318430 036401 382404 669004 577342 404345 580651 293701 915169 547875 375199 627851 092683 014878 593994 698782 188078 > 4²²⁸ [i]