MinT - Best Known (232−99, 232, s)-Nets in Base 4

Best Known (232−99, 232, s)-Nets in Base 4

(232−99, 232, 131)-Net over F₄ — Constructive and digital

Digital (133, 232, 131)-net over F₄, using

1 times m-reduction [i] based on digital (133, 233, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 60, 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, 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]

(232−99, 232, 271)-Net over F₄ — Digital

Digital (133, 232, 271)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−99, 232, 4349)-Net in Base 4 — Upper bound on s

There is no (133, 232, 4350)-net in base 4, because

1 times m-reduction [i] would yield (133, 231, 4350)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 926960 719767 026489 601110 331077 757622 374426 260423 208752 678247 363624 179523 574089 534383 611620 871276 985780 985522 008619 518759 467250 542109 469563 > 4²³¹ [i]