MinT - Best Known (128, 128+81, s)-Nets in Base 4

Best Known (128, 128+81, s)-Nets in Base 4

(128, 128+81, 137)-Net over F₄ — Constructive and digital

Digital (128, 209, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 55, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 154, 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]

(128, 128+81, 337)-Net over F₄ — Digital

Digital (128, 209, 337)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+81, 7069)-Net in Base 4 — Upper bound on s

There is no (128, 209, 7070)-net in base 4, because

1 times m-reduction [i] would yield (128, 208, 7070)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169952 723070 628868 085310 680417 575893 806697 678118 907638 007732 487391 205108 169239 369512 976326 055844 202584 962386 213351 887047 428860 > 4²⁰⁸ [i]