MinT - Best Known (234−90, 234, s)-Nets in Base 4

Best Known (234−90, 234, s)-Nets in Base 4

(234−90, 234, 138)-Net over F₄ — Constructive and digital

Digital (144, 234, 138)-net over F₄, using

10 times m-reduction [i] based on digital (144, 244, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 71, 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, 173, 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]

(234−90, 234, 140)-Net in Base 4 — Constructive

(144, 234, 140)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (26, 71, 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, 163, 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]

(234−90, 234, 382)-Net over F₄ — Digital

Digital (144, 234, 382)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−90, 234, 7902)-Net in Base 4 — Upper bound on s

There is no (144, 234, 7903)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 766 311028 619910 521167 906118 542209 092285 292522 974505 331359 397398 716196 094602 439529 738984 255756 156630 614850 365120 125268 083845 451814 923619 375612 > 4²³⁴ [i]