MinT - Best Known (248−105, 248, s)-Nets in Base 4

Best Known (248−105, 248, s)-Nets in Base 4

(248−105, 248, 137)-Net over F₄ — Constructive and digital

Digital (143, 248, 137)-net over F₄, using

5 times m-reduction [i] based on digital (143, 253, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 70, 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, 183, 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]

(248−105, 248, 294)-Net over F₄ — Digital

Digital (143, 248, 294)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−105, 248, 4838)-Net in Base 4 — Upper bound on s

There is no (143, 248, 4839)-net in base 4, because

1 times m-reduction [i] would yield (143, 247, 4839)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51204 499272 312509 807212 507105 857926 652298 353499 684799 201936 354463 635800 176137 419025 243294 794019 693202 819096 134903 690375 584961 493744 887151 583403 994700 > 4²⁴⁷ [i]