MinT - Best Known (180−109, 180, s)-Nets in Base 4

Best Known (180−109, 180, s)-Nets in Base 4

(180−109, 180, 66)-Net over F₄ — Constructive and digital

Digital (71, 180, 66)-net over F₄, using

t-expansion [i] based on digital (49, 180, 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]

(180−109, 180, 105)-Net over F₄ — Digital

Digital (71, 180, 105)-net over F₄, using

t-expansion [i] based on digital (70, 180, 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]

(180−109, 180, 648)-Net in Base 4 — Upper bound on s

There is no (71, 180, 649)-net in base 4, because

1 times m-reduction [i] would yield (71, 179, 649)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 605258 247301 562946 702627 641424 579234 582402 859052 249139 879081 736614 964186 559895 990206 838785 111436 427365 889504 > 4¹⁷⁹ [i]