MinT - Best Known (68, 68+63, s)-Nets in Base 4

Best Known (68, 68+63, s)-Nets in Base 4

(68, 68+63, 71)-Net over F₄ — Constructive and digital

Digital (68, 131, 71)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 35, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (33, 96, 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]

(68, 68+63, 113)-Net over F₄ — Digital

Digital (68, 131, 113)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+63, 1360)-Net in Base 4 — Upper bound on s

There is no (68, 131, 1361)-net in base 4, because

1 times m-reduction [i] would yield (68, 130, 1361)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 870350 796570 188728 321297 021696 505775 892886 276262 311083 443883 496755 100240 047680 > 4¹³⁰ [i]