MinT - Best Known (258−100, 258, s)-Nets in Base 4

Best Known (258−100, 258, s)-Nets in Base 4

(258−100, 258, 160)-Net over F₄ — Constructive and digital

Digital (158, 258, 160)-net over F₄, using

t-expansion [i] based on digital (157, 258, 160)-net over F₄, using
- 1 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 56)-net over F₄, using
      - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
    2. digital (73, 175, 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]

(258−100, 258, 407)-Net over F₄ — Digital

Digital (158, 258, 407)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−100, 258, 8260)-Net in Base 4 — Upper bound on s

There is no (158, 258, 8261)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214959 410438 824253 579781 450473 897714 471579 050161 750729 468653 000675 578419 233827 693805 032681 772163 339220 540400 832264 929963 376114 698971 230271 130248 928711 673256 > 4²⁵⁸ [i]