MinT - Best Known (258−171, 258, s)-Nets in Base 4

Best Known (258−171, 258, s)-Nets in Base 4

(258−171, 258, 104)-Net over F₄ — Constructive and digital

Digital (87, 258, 104)-net over F₄, using

t-expansion [i] based on digital (73, 258, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(258−171, 258, 129)-Net over F₄ — Digital

Digital (87, 258, 129)-net over F₄, using

t-expansion [i] based on digital (81, 258, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(258−171, 258, 647)-Net in Base 4 — Upper bound on s

There is no (87, 258, 648)-net in base 4, because

1 times m-reduction [i] would yield (87, 257, 648)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 58100 450716 642660 368002 061588 073249 624844 244079 940832 438363 891056 490438 622644 360429 062534 391313 402158 134607 748192 606217 608129 435814 371277 872942 718514 447064 > 4²⁵⁷ [i]