MinT - Best Known (227−143, 227, s)-Nets in Base 4

Best Known (227−143, 227, s)-Nets in Base 4

(227−143, 227, 104)-Net over F₄ — Constructive and digital

Digital (84, 227, 104)-net over F₄, using

t-expansion [i] based on digital (73, 227, 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]

(227−143, 227, 129)-Net over F₄ — Digital

Digital (84, 227, 129)-net over F₄, using

t-expansion [i] based on digital (81, 227, 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]

(227−143, 227, 692)-Net in Base 4 — Upper bound on s

There is no (84, 227, 693)-net in base 4, because

1 times m-reduction [i] would yield (84, 226, 693)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11875 177722 869153 461540 780040 990662 171204 262090 308612 043449 355982 659012 290495 954386 156846 099939 165924 678326 774759 598517 346049 853392 083680 > 4²²⁶ [i]