MinT - Best Known (258−131, 258, s)-Nets in Base 4

Best Known (258−131, 258, s)-Nets in Base 4

(258−131, 258, 130)-Net over F₄ — Constructive and digital

Digital (127, 258, 130)-net over F₄, using

t-expansion [i] based on digital (105, 258, 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]

(258−131, 258, 176)-Net over F₄ — Digital

Digital (127, 258, 176)-net over F₄, using

t-expansion [i] based on digital (125, 258, 176)-net over F₄, using
- net from sequence [i] based on digital (125, 175)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 125 and N(F) ≥ 176, using
    - Drinfeld modules of rank 1 [i]

(258−131, 258, 1951)-Net in Base 4 — Upper bound on s

There is no (127, 258, 1952)-net in base 4, because

1 times m-reduction [i] would yield (127, 257, 1952)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54419 615693 140995 566173 843176 998877 275877 238240 280046 914627 090267 338165 028217 191107 636080 601695 628799 528769 949339 446266 657160 420212 813032 400422 954553 116210 > 4²⁵⁷ [i]