MinT - Best Known (243−92, 243, s)-Nets in Base 4

Best Known (243−92, 243, s)-Nets in Base 4

(243−92, 243, 139)-Net over F₄ — Constructive and digital

Digital (151, 243, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 47, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (104, 196, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 98, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(243−92, 243, 147)-Net in Base 4 — Constructive

(151, 243, 147)-net in base 4, using

4 times m-reduction [i] based on (151, 247, 147)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (30, 78, 43)-net in base 4, using
    - net from sequence [i] based on (30, 42)-sequence in base 4, using
      - base expansion [i] based on digital (60, 42)-sequence over F₂, using
        t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [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]

(243−92, 243, 417)-Net over F₄ — Digital

Digital (151, 243, 417)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−92, 243, 9052)-Net in Base 4 — Upper bound on s

There is no (151, 243, 9053)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 200 745502 322818 949020 355334 644673 826466 087371 049988 175208 377673 601198 049689 582335 069915 487143 423552 904589 379147 943856 860040 050867 849825 164526 242800 > 4²⁴³ [i]