MinT - Best Known (194, 194+45, s)-Nets in Base 2

Best Known (194, 194+45, s)-Nets in Base 2

(194, 194+45, 275)-Net over F₂ — Constructive and digital

Digital (194, 239, 275)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (13, 35, 15)-net over F₂, using
  - net from sequence [i] based on digital (13, 14)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 13 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (159, 204, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 51, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(194, 194+45, 729)-Net over F₂ — Digital

Digital (194, 239, 729)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(194, 194+45, 16314)-Net in Base 2 — Upper bound on s

There is no (194, 239, 16315)-net in base 2, because

1 times m-reduction [i] would yield (194, 238, 16315)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 441718 718325 855662 239802 389405 101764 086040 763952 954526 003925 258962 959994 > 2²³⁸ [i]