MinT - Best Known (77, 77+180, s)-Nets in Base 4

Best Known (77, 77+180, s)-Nets in Base 4

(77, 77+180, 104)-Net over F₄ — Constructive and digital

Digital (77, 257, 104)-net over F₄, using

t-expansion [i] based on digital (73, 257, 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+180, 112)-Net over F₄ — Digital

Digital (77, 257, 112)-net over F₄, using

t-expansion [i] based on digital (73, 257, 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+180, 527)-Net in Base 4 — Upper bound on s

There is no (77, 257, 528)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56145 558948 739430 191443 834249 800144 237854 383904 359765 679700 620497 953558 693010 365169 944540 760554 569572 402872 939292 757447 239223 399775 403046 170812 167312 312672 > 4²⁵⁷ [i]