MinT - Best Known (59−35, 59, s)-Nets in Base 64

Best Known (59−35, 59, s)-Nets in Base 64

(59−35, 59, 242)-Net over F₆₄ — Constructive and digital

Digital (24, 59, 242)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 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, 42, 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]

(59−35, 59, 288)-Net in Base 64 — Constructive

(24, 59, 288)-net in base 64, using

t-expansion [i] based on (22, 59, 288)-net in base 64, using
- 32 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]

(59−35, 59, 344)-Net over F₆₄ — Digital

Digital (24, 59, 344)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁵⁹, 344, F₆₄, 3, 35) (dual of [(344, 3), 973, 36]-NRT-code), using
- constructionÂ X applied to AG(3;F,987P) âŠ‚ AG(3;F,992P) [i] based on
  1. linear OOA(64⁵⁵, 341, F₆₄, 3, 35) (dual of [(341, 3), 968, 36]-NRT-code), using algebraic-geometric NRT-code AG(3;F,987P) [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(64⁵⁰, 341, F₆₄, 3, 30) (dual of [(341, 3), 973, 31]-NRT-code), using algebraic-geometric NRT-code AG(3;F,992P) [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342 (see above)
  3. linear OOA(64⁴, 3, F₆₄, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(64⁴, 64, F₆₄, 3, 4) (dual of [(64, 3), 188, 5]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;188,64) [i]

(59−35, 59, 513)-Net in Base 64

(24, 59, 513)-net in base 64, using

5 times m-reduction [i] based on (24, 64, 513)-net in base 64, using
- base change [i] based on digital (8, 48, 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]

(59−35, 59, 165517)-Net in Base 64 — Upper bound on s

There is no (24, 59, 165518)-net in base 64, because

1 times m-reduction [i] would yield (24, 58, 165518)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 573 401206 446183 148757 355264 059446 036441 874964 505081 899799 262186 006644 421416 772819 864556 904995 428874 436216 > 64⁵⁸ [i]