MinT - Best Known (125, 125+79, s)-Nets in Base 4

Best Known (125, 125+79, s)-Nets in Base 4

(125, 125+79, 134)-Net over F₄ — Constructive and digital

Digital (125, 204, 134)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 52, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 152, 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]

(125, 125+79, 331)-Net over F₄ — Digital

Digital (125, 204, 331)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+79, 6952)-Net in Base 4 — Upper bound on s

There is no (125, 204, 6953)-net in base 4, because

1 times m-reduction [i] would yield (125, 203, 6953)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 166 149058 347182 743228 934566 525147 296482 996809 881670 932666 348099 006928 346734 259167 923963 098376 580011 735426 429922 957265 842888 > 4²⁰³ [i]