MinT - Best Known (137, 137+107, s)-Nets in Base 4

Best Known (137, 137+107, s)-Nets in Base 4

(137, 137+107, 131)-Net over F₄ — Constructive and digital

Digital (137, 244, 131)-net over F₄, using

1 times m-reduction [i] based on digital (137, 245, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 64, 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, 181, 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]

(137, 137+107, 259)-Net over F₄ — Digital

Digital (137, 244, 259)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+107, 3910)-Net in Base 4 — Upper bound on s

There is no (137, 244, 3911)-net in base 4, because

1 times m-reduction [i] would yield (137, 243, 3911)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 200 566683 773081 990113 158381 070957 060665 628054 402958 112831 526928 957373 168798 263895 556575 727505 728058 083247 639415 214065 037018 559982 205022 289046 413640 > 4²⁴³ [i]