MinT - Best Known (166−95, 166, s)-Nets in Base 4

Best Known (166−95, 166, s)-Nets in Base 4

(166−95, 166, 66)-Net over F₄ — Constructive and digital

Digital (71, 166, 66)-net over F₄, using

t-expansion [i] based on digital (49, 166, 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]

(166−95, 166, 105)-Net over F₄ — Digital

Digital (71, 166, 105)-net over F₄, using

t-expansion [i] based on digital (70, 166, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(166−95, 166, 757)-Net in Base 4 — Upper bound on s

There is no (71, 166, 758)-net in base 4, because

1 times m-reduction [i] would yield (71, 165, 758)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2259 766664 387681 418778 409167 645284 290876 382950 437777 431078 015580 593300 792172 251014 970529 423414 247368 > 4¹⁶⁵ [i]