MinT - Best Known (247−103, 247, s)-Nets in Base 4

Best Known (247−103, 247, s)-Nets in Base 4

(247−103, 247, 137)-Net over F₄ — Constructive and digital

Digital (144, 247, 137)-net over F₄, using

9 times m-reduction [i] based on digital (144, 256, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 71, 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, 185, 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]

(247−103, 247, 308)-Net over F₄ — Digital

Digital (144, 247, 308)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−103, 247, 5264)-Net in Base 4 — Upper bound on s

There is no (144, 247, 5265)-net in base 4, because

1 times m-reduction [i] would yield (144, 246, 5265)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12862 157560 533321 187784 648314 974550 812426 303539 681534 801949 100497 785233 785433 984125 294522 350616 016245 336125 561270 202161 799912 017555 888065 401264 305536 > 4²⁴⁶ [i]