MinT - Best Known (138−50, 138, s)-Nets in Base 4

Best Known (138−50, 138, s)-Nets in Base 4

(138−50, 138, 139)-Net over F₄ — Constructive and digital

Digital (88, 138, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 26, 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 (62, 112, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 56, 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]

(138−50, 138, 152)-Net in Base 4 — Constructive

(88, 138, 152)-net in base 4, using

trace code for nets [i] based on (19, 69, 76)-net in base 16, using
- 1 times m-reduction [i] based on (19, 70, 76)-net in base 16, using
  - base change [i] based on digital (5, 56, 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]

(138−50, 138, 298)-Net over F₄ — Digital

Digital (88, 138, 298)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−50, 138, 7122)-Net in Base 4 — Upper bound on s

There is no (88, 138, 7123)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 121583 529169 208048 070383 484096 069353 239741 517890 195325 657038 677429 372527 245468 019056 > 4¹³⁸ [i]