MinT - Best Known (225−115, 225, s)-Nets in Base 4

Best Known (225−115, 225, s)-Nets in Base 4

(225−115, 225, 130)-Net over F₄ — Constructive and digital

Digital (110, 225, 130)-net over F₄, using

t-expansion [i] based on digital (105, 225, 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]

(225−115, 225, 165)-Net over F₄ — Digital

Digital (110, 225, 165)-net over F₄, using

t-expansion [i] based on digital (109, 225, 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]

(225−115, 225, 1662)-Net in Base 4 — Upper bound on s

There is no (110, 225, 1663)-net in base 4, because

1 times m-reduction [i] would yield (110, 224, 1663)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 727 607247 385920 199523 229424 409797 269107 031619 304559 229768 340405 489581 486257 987879 097891 295573 774110 521267 902819 703434 408917 560018 239710 > 4²²⁴ [i]