MinT - Best Known (77, 77+135, s)-Nets in Base 4

Best Known (77, 77+135, s)-Nets in Base 4

(77, 77+135, 104)-Net over F₄ — Constructive and digital

Digital (77, 212, 104)-net over F₄, using

t-expansion [i] based on digital (73, 212, 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]

(77, 77+135, 112)-Net over F₄ — Digital

Digital (77, 212, 112)-net over F₄, using

t-expansion [i] based on digital (73, 212, 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]

(77, 77+135, 622)-Net in Base 4 — Upper bound on s

There is no (77, 212, 623)-net in base 4, because

1 times m-reduction [i] would yield (77, 211, 623)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 982864 585787 371732 065242 577821 827388 228296 894446 350135 478425 484962 510972 636487 581135 913029 795522 855122 637710 320166 697495 569848 > 4²¹¹ [i]