MinT - Best Known (243−129, 243, s)-Nets in Base 4

Best Known (243−129, 243, s)-Nets in Base 4

(243−129, 243, 130)-Net over F₄ — Constructive and digital

Digital (114, 243, 130)-net over F₄, using

t-expansion [i] based on digital (105, 243, 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]

(243−129, 243, 165)-Net over F₄ — Digital

Digital (114, 243, 165)-net over F₄, using

t-expansion [i] based on digital (109, 243, 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]

(243−129, 243, 1502)-Net in Base 4 — Upper bound on s

There is no (114, 243, 1503)-net in base 4, because

1 times m-reduction [i] would yield (114, 242, 1503)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 809541 663304 339229 754416 620879 835618 537871 180206 587013 496127 687181 319458 601330 779402 339934 598248 223232 246753 678747 754512 295889 242884 265282 383831 > 4²⁴² [i]