MinT - Best Known (259−133, 259, s)-Nets in Base 4

Best Known (259−133, 259, s)-Nets in Base 4

(259−133, 259, 130)-Net over F₄ — Constructive and digital

Digital (126, 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]

(259−133, 259, 176)-Net over F₄ — Digital

Digital (126, 259, 176)-net over F₄, using

t-expansion [i] based on digital (125, 259, 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]

(259−133, 259, 1858)-Net in Base 4 — Upper bound on s

There is no (126, 259, 1859)-net in base 4, because

1 times m-reduction [i] would yield (126, 258, 1859)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 221847 655968 780159 969921 280702 021261 610441 171340 206145 021504 767931 007603 466341 934344 021697 971427 672071 289857 846683 898685 796059 168099 269786 264421 218331 586589 > 4²⁵⁸ [i]