MinT - Best Known (127−44, 127, s)-Nets in Base 4

Best Known (127−44, 127, s)-Nets in Base 4

(127−44, 127, 147)-Net over F₄ — Constructive and digital

Digital (83, 127, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 27, 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 (56, 100, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 50, 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]

(127−44, 127, 152)-Net in Base 4 — Constructive

(83, 127, 152)-net in base 4, using

3 times m-reduction [i] based on (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]

(127−44, 127, 326)-Net over F₄ — Digital

Digital (83, 127, 326)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−44, 127, 9003)-Net in Base 4 — Upper bound on s

There is no (83, 127, 9004)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 28998 277925 704255 085225 474966 368868 514490 650956 471451 686663 436587 573512 203940 > 4¹²⁷ [i]