MinT - Best Known (128, 128+77, s)-Nets in Base 4

Best Known (128, 128+77, s)-Nets in Base 4

(128, 128+77, 139)-Net over F₄ — Constructive and digital

Digital (128, 205, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 39, 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 (89, 166, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 83, 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]

(128, 128+77, 369)-Net over F₄ — Digital

Digital (128, 205, 369)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+77, 8515)-Net in Base 4 — Upper bound on s

There is no (128, 205, 8516)-net in base 4, because

1 times m-reduction [i] would yield (128, 204, 8516)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 662 591359 954725 850814 872225 195683 413840 577897 620296 234936 983768 621673 055545 121817 288922 023035 118610 300380 768699 563964 052240 > 4²⁰⁴ [i]