MinT - Best Known (89, 89+149, s)-Nets in Base 4

Best Known (89, 89+149, s)-Nets in Base 4

(89, 89+149, 104)-Net over F₄ — Constructive and digital

Digital (89, 238, 104)-net over F₄, using

t-expansion [i] based on digital (73, 238, 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]

(89, 89+149, 129)-Net over F₄ — Digital

Digital (89, 238, 129)-net over F₄, using

t-expansion [i] based on digital (81, 238, 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]

(89, 89+149, 742)-Net in Base 4 — Upper bound on s

There is no (89, 238, 743)-net in base 4, because

1 times m-reduction [i] would yield (89, 237, 743)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51831 020868 503359 237892 568874 794279 972038 596353 065555 859935 905336 700249 627352 085567 262119 174601 432857 709257 367348 079444 602342 934314 938513 143724 > 4²³⁷ [i]