MinT - Best Known (77, 77+139, s)-Nets in Base 4

Best Known (77, 77+139, s)-Nets in Base 4

(77, 77+139, 104)-Net over F₄ — Constructive and digital

Digital (77, 216, 104)-net over F₄, using

t-expansion [i] based on digital (73, 216, 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+139, 112)-Net over F₄ — Digital

Digital (77, 216, 112)-net over F₄, using

t-expansion [i] based on digital (73, 216, 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+139, 609)-Net in Base 4 — Upper bound on s

There is no (77, 216, 610)-net in base 4, because

1 times m-reduction [i] would yield (77, 215, 610)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2967 991986 787970 569172 697571 814453 768986 399885 956725 422827 551478 963569 900608 977111 987129 572915 004282 241451 836309 515914 940387 480088 > 4²¹⁵ [i]