MinT - Best Known (250−103, 250, s)-Nets in Base 4

Best Known (250−103, 250, s)-Nets in Base 4

(250−103, 250, 138)-Net over F₄ — Constructive and digital

Digital (147, 250, 138)-net over F₄, using

3 times m-reduction [i] based on digital (147, 253, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 74, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 179, 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]

(250−103, 250, 324)-Net over F₄ — Digital

Digital (147, 250, 324)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−103, 250, 5715)-Net in Base 4 — Upper bound on s

There is no (147, 250, 5716)-net in base 4, because

1 times m-reduction [i] would yield (147, 249, 5716)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 824032 721437 092920 937899 364208 237162 122560 161150 678449 503627 759735 111298 196668 919113 788724 896316 868331 837071 368785 137860 055327 219996 600298 280310 024510 > 4²⁴⁹ [i]