MinT - Best Known (129, 129+83, s)-Nets in Base 4

Best Known (129, 129+83, s)-Nets in Base 4

(129, 129+83, 137)-Net over F₄ — Constructive and digital

Digital (129, 212, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 56, 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, 156, 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]

(129, 129+83, 330)-Net over F₄ — Digital

Digital (129, 212, 330)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+83, 6714)-Net in Base 4 — Upper bound on s

There is no (129, 212, 6715)-net in base 4, because

1 times m-reduction [i] would yield (129, 211, 6715)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 847014 248617 079392 049886 008167 629293 138857 477468 565547 771549 999767 290760 326971 234095 812850 518100 843990 097212 130418 900129 508850 > 4²¹¹ [i]