MinT - Best Known (232−89, 232, s)-Nets in Base 4

Best Known (232−89, 232, s)-Nets in Base 4

(232−89, 232, 138)-Net over F₄ — Constructive and digital

Digital (143, 232, 138)-net over F₄, using

9 times m-reduction [i] based on digital (143, 241, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 70, 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, 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]

(232−89, 232, 140)-Net in Base 4 — Constructive

(143, 232, 140)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (26, 70, 36)-net in base 4, using
  - net from sequence [i] based on (26, 35)-sequence in base 4, using
    - base expansion [i] based on digital (52, 35)-sequence over F₂, using
      - t-expansion [i] based on digital (51, 35)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
2. digital (73, 162, 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]

(232−89, 232, 383)-Net over F₄ — Digital

Digital (143, 232, 383)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−89, 232, 8293)-Net in Base 4 — Upper bound on s

There is no (143, 232, 8294)-net in base 4, because

1 times m-reduction [i] would yield (143, 231, 8294)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 949798 669536 259100 508512 916698 335249 427825 730651 169222 964770 151819 722513 164786 648721 872605 315663 328661 826125 373720 924045 608058 316818 681336 > 4²³¹ [i]