MinT - Best Known (109, 252, s)-Nets in Base 4

Best Known (109, 252, s)-Nets in Base 4

(109, 252, 130)-Net over F₄ — Constructive and digital

Digital (109, 252, 130)-net over F₄, using

t-expansion [i] based on digital (105, 252, 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]

(109, 252, 165)-Net over F₄ — Digital

Digital (109, 252, 165)-net over F₄, using

net from sequence [i] based on digital (109, 164)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
  - Drinfeld modules of rank 1 [i]

(109, 252, 1163)-Net in Base 4 — Upper bound on s

There is no (109, 252, 1164)-net in base 4, because

1 times m-reduction [i] would yield (109, 251, 1164)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 132747 476304 418968 383670 075863 014249 695067 354454 966671 422893 289006 900262 163104 845861 308790 973102 043798 349288 430013 122655 929905 355247 434774 367301 670500 > 4²⁵¹ [i]