MinT - Best Known (139, 139+109, s)-Nets in Base 4

Best Known (139, 139+109, s)-Nets in Base 4

(139, 139+109, 132)-Net over F₄ — Constructive and digital

Digital (139, 248, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 66, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 182, 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]

(139, 139+109, 261)-Net over F₄ — Digital

Digital (139, 248, 261)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+109, 3921)-Net in Base 4 — Upper bound on s

There is no (139, 248, 3922)-net in base 4, because

1 times m-reduction [i] would yield (139, 247, 3922)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51773 816107 191365 151549 935065 666931 342927 661680 698175 783880 113390 656309 393583 156219 906243 360367 897734 949910 306626 155691 863153 341038 614653 011824 111200 > 4²⁴⁷ [i]