MinT - Best Known (178−99, 178, s)-Nets in Base 4

Best Known (178−99, 178, s)-Nets in Base 4

(178−99, 178, 104)-Net over F₄ — Constructive and digital

Digital (79, 178, 104)-net over F₄, using

t-expansion [i] based on digital (73, 178, 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]

(178−99, 178, 112)-Net over F₄ — Digital

Digital (79, 178, 112)-net over F₄, using

t-expansion [i] based on digital (73, 178, 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]

(178−99, 178, 913)-Net in Base 4 — Upper bound on s

There is no (79, 178, 914)-net in base 4, because

1 times m-reduction [i] would yield (79, 177, 914)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38538 890577 482859 712399 250952 768408 098843 702798 202274 208297 662207 658148 249472 796109 986378 804075 399514 327748 > 4¹⁷⁷ [i]