MinT - Best Known (244−139, 244, s)-Nets in Base 4

Best Known (244−139, 244, s)-Nets in Base 4

(244−139, 244, 130)-Net over F₄ — Constructive and digital

Digital (105, 244, 130)-net over F₄, using

net from sequence [i] based on digital (105, 129)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
  - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(244−139, 244, 144)-Net over F₄ — Digital

Digital (105, 244, 144)-net over F₄, using

t-expansion [i] based on digital (91, 244, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(244−139, 244, 1110)-Net in Base 4 — Upper bound on s

There is no (105, 244, 1111)-net in base 4, because

1 times m-reduction [i] would yield (105, 243, 1111)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 206 546809 202440 178010 799495 886706 553263 256561 465966 236641 709817 363010 958118 331725 476456 640078 618314 535770 792712 154644 250347 572662 001094 235453 017630 > 4²⁴³ [i]