MinT - Best Known (138, 138+91, s)-Nets in Base 4

Best Known (138, 138+91, s)-Nets in Base 4

(138, 138+91, 137)-Net over F₄ — Constructive and digital

Digital (138, 229, 137)-net over F₄, using

9 times m-reduction [i] based on digital (138, 238, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 65, 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, 173, 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]

(138, 138+91, 337)-Net over F₄ — Digital

Digital (138, 229, 337)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+91, 6562)-Net in Base 4 — Upper bound on s

There is no (138, 229, 6563)-net in base 4, because

1 times m-reduction [i] would yield (138, 228, 6563)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 187019 617446 550100 451803 836543 593305 406928 488330 895071 434564 341200 001143 477156 045439 323781 170573 833625 946840 999076 799180 428751 519365 425600 > 4²²⁸ [i]