MinT - Best Known (258−165, 258, s)-Nets in Base 4

Best Known (258−165, 258, s)-Nets in Base 4

(258−165, 258, 104)-Net over F₄ — Constructive and digital

Digital (93, 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]

(258−165, 258, 144)-Net over F₄ — Digital

Digital (93, 258, 144)-net over F₄, using

t-expansion [i] based on digital (91, 258, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(258−165, 258, 739)-Net in Base 4 — Upper bound on s

There is no (93, 258, 740)-net in base 4, because

1 times m-reduction [i] would yield (93, 257, 740)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56914 632419 738818 522807 136392 510319 470813 155828 889436 154869 511269 936142 823809 305872 801393 921546 739239 136394 411328 874335 351381 976833 965940 717526 355145 941970 > 4²⁵⁷ [i]