MinT - Best Known (211−133, 211, s)-Nets in Base 4

Best Known (211−133, 211, s)-Nets in Base 4

(211−133, 211, 104)-Net over F₄ — Constructive and digital

Digital (78, 211, 104)-net over F₄, using

t-expansion [i] based on digital (73, 211, 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]

(211−133, 211, 112)-Net over F₄ — Digital

Digital (78, 211, 112)-net over F₄, using

t-expansion [i] based on digital (73, 211, 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]

(211−133, 211, 644)-Net in Base 4 — Upper bound on s

There is no (78, 211, 645)-net in base 4, because

1 times m-reduction [i] would yield (78, 210, 645)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 793023 727907 509124 626838 483145 484714 391184 899321 789939 797746 234839 999999 239867 937705 104542 679522 774771 408439 517727 494386 464052 > 4²¹⁰ [i]