MinT - Best Known (258−98, 258, s)-Nets in Base 4

Best Known (258−98, 258, s)-Nets in Base 4

(258−98, 258, 160)-Net over F₄ — Constructive and digital

Digital (160, 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]

(258−98, 258, 435)-Net over F₄ — Digital

Digital (160, 258, 435)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−98, 258, 9383)-Net in Base 4 — Upper bound on s

There is no (160, 258, 9384)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 215262 833994 190753 182398 400171 853532 966769 876883 568094 275185 523591 721099 378066 431817 414549 205430 294574 223815 287827 048207 253046 829024 633241 981601 336200 513224 > 4²⁵⁸ [i]