MinT - Best Known (222−137, 222, s)-Nets in Base 4

Best Known (222−137, 222, s)-Nets in Base 4

(222−137, 222, 104)-Net over F₄ — Constructive and digital

Digital (85, 222, 104)-net over F₄, using

t-expansion [i] based on digital (73, 222, 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]

(222−137, 222, 129)-Net over F₄ — Digital

Digital (85, 222, 129)-net over F₄, using

t-expansion [i] based on digital (81, 222, 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]

(222−137, 222, 734)-Net in Base 4 — Upper bound on s

There is no (85, 222, 735)-net in base 4, because

1 times m-reduction [i] would yield (85, 221, 735)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 942038 647131 431260 519271 976923 585406 857829 342927 985280 487261 885556 241186 204211 880133 126592 822939 479596 945291 952388 541714 075193 281703 > 4²²¹ [i]