MinT - Best Known (244−95, 244, s)-Nets in Base 4

Best Known (244−95, 244, s)-Nets in Base 4

(244−95, 244, 138)-Net over F₄ — Constructive and digital

Digital (149, 244, 138)-net over F₄, using

15 times m-reduction [i] based on digital (149, 259, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 76, 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, 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]

(244−95, 244, 146)-Net in Base 4 — Constructive

(149, 244, 146)-net in base 4, using

3 times m-reduction [i] based on (149, 247, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 76, 42)-net in base 4, using
    - net from sequence [i] based on (27, 41)-sequence in base 4, using
      - base expansion [i] based on digital (54, 41)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 171, 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]

(244−95, 244, 381)-Net over F₄ — Digital

Digital (149, 244, 381)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−95, 244, 7900)-Net in Base 4 — Upper bound on s

There is no (149, 244, 7901)-net in base 4, because

1 times m-reduction [i] would yield (149, 243, 7901)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 200 702704 230638 788346 374435 126047 658510 177971 383316 107346 306300 794130 884688 810484 174459 306876 942292 740775 244838 706595 885721 532507 309128 099244 703712 > 4²⁴³ [i]