MinT - Best Known (168−99, 168, s)-Nets in Base 4

Best Known (168−99, 168, s)-Nets in Base 4

(168−99, 168, 66)-Net over F₄ — Constructive and digital

Digital (69, 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]

(168−99, 168, 99)-Net over F₄ — Digital

Digital (69, 168, 99)-net over F₄, using

t-expansion [i] based on digital (61, 168, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(168−99, 168, 678)-Net in Base 4 — Upper bound on s

There is no (69, 168, 679)-net in base 4, because

1 times m-reduction [i] would yield (69, 167, 679)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36266 495387 152028 473675 689230 744578 538909 772992 358482 037006 192093 316074 695342 536891 852227 740837 860720 > 4¹⁶⁷ [i]