MinT - Best Known (81, 81+43, s)-Nets in Base 4

Best Known (81, 81+43, s)-Nets in Base 4

(81, 81+43, 147)-Net over F₄ — Constructive and digital

Digital (81, 124, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 26, 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 (55, 98, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 49, 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]

(81, 81+43, 152)-Net in Base 4 — Constructive

(81, 124, 152)-net in base 4, using

2 times m-reduction [i] based on (81, 126, 152)-net in base 4, using
- trace code for nets [i] based on (18, 63, 76)-net in base 16, using
  - 2 times m-reduction [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]

(81, 81+43, 318)-Net over F₄ — Digital

Digital (81, 124, 318)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+43, 9704)-Net in Base 4 — Upper bound on s

There is no (81, 124, 9705)-net in base 4, because

1 times m-reduction [i] would yield (81, 123, 9705)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 113 255583 984456 083917 926379 086728 607047 905866 825368 982280 963569 173552 734336 > 4¹²³ [i]