MinT - Best Known (77, 77+163, s)-Nets in Base 4

Best Known (77, 77+163, s)-Nets in Base 4

(77, 77+163, 104)-Net over F₄ — Constructive and digital

Digital (77, 240, 104)-net over F₄, using

t-expansion [i] based on digital (73, 240, 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+163, 112)-Net over F₄ — Digital

Digital (77, 240, 112)-net over F₄, using

t-expansion [i] based on digital (73, 240, 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+163, 552)-Net in Base 4 — Upper bound on s

There is no (77, 240, 553)-net in base 4, because

1 times m-reduction [i] would yield (77, 239, 553)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 828102 303984 022113 752523 472634 066376 360278 154263 961754 517780 347447 025391 540760 791144 530498 543939 119088 452036 418516 081035 156019 278232 384707 566400 > 4²³⁹ [i]