MinT - Best Known (130, 209, s)-Nets in Base 4

Best Known (130, 209, s)-Nets in Base 4

(130, 209, 137)-Net over F₄ — Constructive and digital

Digital (130, 209, 137)-net over F₄, using

5 times m-reduction [i] based on digital (130, 214, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 57, 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, 157, 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]

(130, 209, 367)-Net over F₄ — Digital

Digital (130, 209, 367)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(130, 209, 8310)-Net in Base 4 — Upper bound on s

There is no (130, 209, 8311)-net in base 4, because

1 times m-reduction [i] would yield (130, 208, 8311)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169674 275487 093974 455976 343382 351209 541121 010228 883551 160368 069045 933398 015702 632982 638139 868213 153215 474526 434430 789897 944380 > 4²⁰⁸ [i]