MinT - Best Known (128, 206, s)-Nets in Base 4

Best Known (128, 206, s)-Nets in Base 4

(128, 206, 137)-Net over F₄ — Constructive and digital

Digital (128, 206, 137)-net over F₄, using

2 times m-reduction [i] based on digital (128, 208, 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, 153, 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, 206, 361)-Net over F₄ — Digital

Digital (128, 206, 361)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 206, 7737)-Net in Base 4 — Upper bound on s

There is no (128, 206, 7738)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10579 220548 516940 286234 942040 606869 871601 474243 703216 072946 620014 032171 171271 898827 943018 549727 662819 895270 522536 195596 759308 > 4²⁰⁶ [i]