MinT - Best Known (145, 145+88, s)-Nets in Base 4

Best Known (145, 145+88, s)-Nets in Base 4

(145, 145+88, 139)-Net over F₄ — Constructive and digital

Digital (145, 233, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 45, 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 (100, 188, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 94, 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]

(145, 145+88, 146)-Net in Base 4 — Constructive

(145, 233, 146)-net in base 4, using

2 times m-reduction [i] based on (145, 235, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 72, 42)-net in base 4, using
    - net from sequence [i] based on (27, 41)-sequence in base 4, using
      - base expansion [i] based on digital (54, 41)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 163, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(145, 145+88, 405)-Net over F₄ — Digital

Digital (145, 233, 405)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(145, 145+88, 8835)-Net in Base 4 — Upper bound on s

There is no (145, 233, 8836)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 191 390737 811398 320701 724848 770425 718545 355588 930253 114600 344210 961569 031238 843324 793489 740546 214189 003927 783251 499652 280257 188856 064704 239840 > 4²³³ [i]