MinT - Best Known (68, 68+132, s)-Nets in Base 4

Best Known (68, 68+132, s)-Nets in Base 4

(68, 68+132, 66)-Net over F₄ — Constructive and digital

Digital (68, 200, 66)-net over F₄, using

t-expansion [i] based on digital (49, 200, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(68, 68+132, 99)-Net over F₄ — Digital

Digital (68, 200, 99)-net over F₄, using

t-expansion [i] based on digital (61, 200, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(68, 68+132, 512)-Net in Base 4 — Upper bound on s

There is no (68, 200, 513)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 625887 850687 044106 244566 802109 572637 842849 291935 405888 486407 395123 300693 236181 323410 289186 247369 651657 460884 704534 574720 > 4²⁰⁰ [i]