MinT - Best Known (259−153, 259, s)-Nets in Base 4

Best Known (259−153, 259, s)-Nets in Base 4

(259−153, 259, 130)-Net over F₄ — Constructive and digital

Digital (106, 259, 130)-net over F₄, using

t-expansion [i] based on digital (105, 259, 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]

(259−153, 259, 144)-Net over F₄ — Digital

Digital (106, 259, 144)-net over F₄, using

t-expansion [i] based on digital (91, 259, 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]

(259−153, 259, 1012)-Net in Base 4 — Upper bound on s

There is no (106, 259, 1013)-net in base 4, because

1 times m-reduction [i] would yield (106, 258, 1013)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 227329 193355 839055 675877 576176 610566 483301 518599 939201 378728 413529 391719 019255 229743 792008 340024 469483 304886 886452 434025 783258 781776 199981 251261 680651 739110 > 4²⁵⁸ [i]