MinT - Best Known (168−97, 168, s)-Nets in Base 4

Best Known (168−97, 168, s)-Nets in Base 4

(168−97, 168, 66)-Net over F₄ — Constructive and digital

Digital (71, 168, 66)-net over F₄, using

t-expansion [i] based on digital (49, 168, 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]

(168−97, 168, 105)-Net over F₄ — Digital

Digital (71, 168, 105)-net over F₄, using

t-expansion [i] based on digital (70, 168, 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]

(168−97, 168, 738)-Net in Base 4 — Upper bound on s

There is no (71, 168, 739)-net in base 4, because

1 times m-reduction [i] would yield (71, 167, 739)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 37007 710593 006045 688876 368851 359655 060941 743424 927998 698755 103002 812593 308347 827220 052546 641667 585252 > 4¹⁶⁷ [i]