MinT - Best Known (127, 127+77, s)-Nets in Base 4

Best Known (127, 127+77, s)-Nets in Base 4

(127, 127+77, 137)-Net over F₄ — Constructive and digital

Digital (127, 204, 137)-net over F₄, using

1 times m-reduction [i] based on digital (127, 205, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 54, 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, 151, 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]

(127, 127+77, 362)-Net over F₄ — Digital

Digital (127, 204, 362)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+77, 8209)-Net in Base 4 — Upper bound on s

There is no (127, 204, 8210)-net in base 4, because

1 times m-reduction [i] would yield (127, 203, 8210)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 793445 518021 236594 771849 450820 815135 196551 645863 331201 021946 041206 793772 532450 314362 584426 029105 930673 018859 757315 287040 > 4²⁰³ [i]