MinT - Best Known (241−99, 241, s)-Nets in Base 4

Best Known (241−99, 241, s)-Nets in Base 4

(241−99, 241, 137)-Net over F₄ — Constructive and digital

Digital (142, 241, 137)-net over F₄, using

9 times m-reduction [i] based on digital (142, 250, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 69, 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, 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]

(241−99, 241, 317)-Net over F₄ — Digital

Digital (142, 241, 317)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−99, 241, 5622)-Net in Base 4 — Upper bound on s

There is no (142, 241, 5623)-net in base 4, because

1 times m-reduction [i] would yield (142, 240, 5623)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 125785 252036 139664 495630 622224 096873 771002 147285 398170 368450 510136 035299 928190 388164 454248 738454 166584 377638 597499 507699 567616 612638 243304 932466 > 4²⁴⁰ [i]