MinT - Best Known (77, 77+40, s)-Nets in Base 4

Best Known (77, 77+40, s)-Nets in Base 4

(77, 77+40, 147)-Net over F₄ — Constructive and digital

Digital (77, 117, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 25, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (52, 92, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 46, 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]

(77, 77+40, 152)-Net in Base 4 — Constructive

(77, 117, 152)-net in base 4, using

3 times m-reduction [i] based on (77, 120, 152)-net in base 4, using
- trace code for nets [i] based on (17, 60, 76)-net in base 16, using
  - base change [i] based on digital (5, 48, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(77, 77+40, 320)-Net over F₄ — Digital

Digital (77, 117, 320)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+40, 9193)-Net in Base 4 — Upper bound on s

There is no (77, 117, 9194)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 27646 485002 875142 797542 012448 268667 159421 424978 796400 667366 942822 573884 > 4¹¹⁷ [i]