MinT - Best Known (244−96, 244, s)-Nets in Base 4

Best Known (244−96, 244, s)-Nets in Base 4

(244−96, 244, 138)-Net over F₄ — Constructive and digital

Digital (148, 244, 138)-net over F₄, using

12 times m-reduction [i] based on digital (148, 256, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 75, 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, 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]

(244−96, 244, 146)-Net in Base 4 — Constructive

(148, 244, 146)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (27, 75, 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, 169, 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]

(244−96, 244, 368)-Net over F₄ — Digital

Digital (148, 244, 368)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−96, 244, 7140)-Net in Base 4 — Upper bound on s

There is no (148, 244, 7141)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 800 010364 940773 814034 223605 258740 852136 204750 658693 375579 917304 589538 049700 439868 404672 527548 683461 815395 417229 446734 482441 313877 560700 248678 677924 > 4²⁴⁴ [i]