MinT - Best Known (249−163, 249, s)-Nets in Base 4

Best Known (249−163, 249, s)-Nets in Base 4

(249−163, 249, 104)-Net over F₄ — Constructive and digital

Digital (86, 249, 104)-net over F₄, using

t-expansion [i] based on digital (73, 249, 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]

(249−163, 249, 129)-Net over F₄ — Digital

Digital (86, 249, 129)-net over F₄, using

t-expansion [i] based on digital (81, 249, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(249−163, 249, 654)-Net in Base 4 — Upper bound on s

There is no (86, 249, 655)-net in base 4, because

1 times m-reduction [i] would yield (86, 248, 655)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 206262 529664 298783 975725 616707 859077 913112 008355 244964 839450 257900 471617 170737 173370 842430 619828 371753 624411 060841 777120 576396 480147 477834 654514 810832 > 4²⁴⁸ [i]