MinT - Best Known (123−43, 123, s)-Nets in Base 4

Best Known (123−43, 123, s)-Nets in Base 4

(123−43, 123, 145)-Net over F₄ — Constructive and digital

Digital (80, 123, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 25, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, 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]

(123−43, 123, 152)-Net in Base 4 — Constructive

(80, 123, 152)-net in base 4, using

1 times m-reduction [i] based on (80, 124, 152)-net in base 4, using
- trace code for nets [i] based on (18, 62, 76)-net in base 16, using
  - 3 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]

(123−43, 123, 307)-Net over F₄ — Digital

Digital (80, 123, 307)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−43, 123, 9083)-Net in Base 4 — Upper bound on s

There is no (80, 123, 9084)-net in base 4, because

1 times m-reduction [i] would yield (80, 122, 9084)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 28 317933 367031 357552 611156 148082 748646 870919 934449 613388 122987 250278 913722 > 4¹²² [i]