MinT - Best Known (244−165, 244, s)-Nets in Base 4

Best Known (244−165, 244, s)-Nets in Base 4

(244−165, 244, 104)-Net over F₄ — Constructive and digital

Digital (79, 244, 104)-net over F₄, using

t-expansion [i] based on digital (73, 244, 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]

(244−165, 244, 112)-Net over F₄ — Digital

Digital (79, 244, 112)-net over F₄, using

t-expansion [i] based on digital (73, 244, 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]

(244−165, 244, 570)-Net in Base 4 — Upper bound on s

There is no (79, 244, 571)-net in base 4, because

1 times m-reduction [i] would yield (79, 243, 571)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 220 153516 866649 088488 999516 309302 640038 583587 775220 845262 262656 509820 691815 226291 120187 596727 363434 618432 571266 449598 886779 131765 424253 224058 806760 > 4²⁴³ [i]