MinT - Best Known (83, 83+47, s)-Nets in Base 4

Best Known (83, 83+47, s)-Nets in Base 4

(83, 83+47, 139)-Net over F₄ — Constructive and digital

Digital (83, 130, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 24, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (59, 106, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 53, 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]

(83, 83+47, 152)-Net in Base 4 — Constructive

(83, 130, 152)-net in base 4, using

trace code for nets [i] based on (18, 65, 76)-net in base 16, using
- base change [i] based on digital (5, 52, 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]

(83, 83+47, 285)-Net over F₄ — Digital

Digital (83, 130, 285)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(83, 83+47, 7464)-Net in Base 4 — Upper bound on s

There is no (83, 130, 7465)-net in base 4, because

1 times m-reduction [i] would yield (83, 129, 7465)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 463342 877069 540894 383780 625704 959344 523084 377602 676627 226932 780924 092609 396336 > 4¹²⁹ [i]