MinT - Best Known (55−32, 55, s)-Nets in Base 64

Best Known (55−32, 55, s)-Nets in Base 64

(55−32, 55, 242)-Net over F₆₄ — Constructive and digital

Digital (23, 55, 242)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 16, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 39, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(55−32, 55, 288)-Net in Base 64 — Constructive

(23, 55, 288)-net in base 64, using

t-expansion [i] based on (22, 55, 288)-net in base 64, using
- 36 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
  - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(55−32, 55, 344)-Net over F₆₄ — Digital

Digital (23, 55, 344)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁵⁵, 344, F₆₄, 2, 32) (dual of [(344, 2), 633, 33]-NRT-code), using
- constructionÂ X applied to AG(2;F,649P) âŠ‚ AG(2;F,653P) [i] based on
  1. linear OOA(64⁵², 341, F₆₄, 2, 32) (dual of [(341, 2), 630, 33]-NRT-code), using algebraic-geometric NRT-code AG(2;F,649P) [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
    - a curve from the manYPoints database [i]
  2. linear OOA(64⁴⁸, 341, F₆₄, 2, 28) (dual of [(341, 2), 634, 29]-NRT-code), using algebraic-geometric NRT-code AG(2;F,653P) [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342 (see above)
  3. linear OOA(64³, 3, F₆₄, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(64³, 64, F₆₄, 2, 3) (dual of [(64, 2), 125, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;125,64) [i]

(55−32, 55, 513)-Net in Base 64

(23, 55, 513)-net in base 64, using

5 times m-reduction [i] based on (23, 60, 513)-net in base 64, using
- base change [i] based on digital (8, 45, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(55−32, 55, 174551)-Net in Base 64 — Upper bound on s

There is no (23, 55, 174552)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2187 434878 989981 093127 855709 859224 412542 424282 533892 261342 583060 352060 283952 621380 449854 135426 371433 > 64⁵⁵ [i]