MinT - Best Known (258−115, 258, s)-Nets in Base 4

Best Known (258−115, 258, s)-Nets in Base 4

(258−115, 258, 134)-Net over F₄ — Constructive and digital

Digital (143, 258, 134)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 70, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 188, 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]

(258−115, 258, 259)-Net over F₄ — Digital

Digital (143, 258, 259)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−115, 258, 3767)-Net in Base 4 — Upper bound on s

There is no (143, 258, 3768)-net in base 4, because

1 times m-reduction [i] would yield (143, 257, 3768)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53942 317119 068273 296111 881226 292236 913383 515536 059309 849569 604127 304606 114309 418259 054874 681716 988306 035591 369245 044029 996887 223793 675263 954674 864463 908144 > 4²⁵⁷ [i]