MinT - Best Known (240−105, 240, s)-Nets in Base 4

Best Known (240−105, 240, s)-Nets in Base 4

(240−105, 240, 131)-Net over F₄ — Constructive and digital

Digital (135, 240, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 62, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 178, 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]

(240−105, 240, 257)-Net over F₄ — Digital

Digital (135, 240, 257)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−105, 240, 3901)-Net in Base 4 — Upper bound on s

There is no (135, 240, 3902)-net in base 4, because

1 times m-reduction [i] would yield (135, 239, 3902)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 787289 280712 478104 047841 866426 443750 160013 482859 402238 692509 846462 343145 773535 305346 275283 883957 970461 490484 557887 072010 717817 500239 491071 096825 > 4²³⁹ [i]