MinT - Best Known (128, 128+89, s)-Nets in Base 4

Best Known (128, 128+89, s)-Nets in Base 4

(128, 128+89, 131)-Net over F₄ — Constructive and digital

Digital (128, 217, 131)-net over F₄, using

1 times m-reduction [i] based on digital (128, 218, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 55, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 163, 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+89, 290)-Net over F₄ — Digital

Digital (128, 217, 290)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+89, 5156)-Net in Base 4 — Upper bound on s

There is no (128, 217, 5157)-net in base 4, because

1 times m-reduction [i] would yield (128, 216, 5157)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11153 254759 348784 785348 906706 049974 143074 064760 718323 976667 995131 663492 590048 781061 685699 518335 833771 048993 324081 613851 581811 905664 > 4²¹⁶ [i]