MinT - Best Known (36−21, 36, s)-Nets in Base 64

Best Known (36−21, 36, s)-Nets in Base 64

(36−21, 36, 193)-Net over F₆₄ — Constructive and digital

Digital (15, 36, 193)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 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 (5, 26, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(36−21, 36, 260)-Net over F₆₄ — Digital

Digital (15, 36, 260)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁶, 260, F₆₄, 21) (dual of [260, 224, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(64³⁶, 263, F₆₄, 21) (dual of [263, 227, 22]-code), using
  - constructionÂ X applied to AG(F,234P) âŠ‚ AG(F,238P) [i] based on
    1. linear OA(64³³, 256, F₆₄, 21) (dual of [256, 223, 22]-code), using algebraic-geometric code AG(F,234P) [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
      - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]
    2. linear OA(64²⁹, 256, F₆₄, 17) (dual of [256, 227, 18]-code), using algebraic-geometric code AG(F,238P) [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257 (see above)
    3. linear OA(64³, 7, F₆₄, 3) (dual of [7, 4, 4]-code or 7-arc in PG(2,64) or 7-cap in PG(2,64)), using
      - discarding factors / shortening the dual code based on linear OA(64³, 64, F₆₄, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
        Reedâ€“Solomon code RS(61,64) [i]

(36−21, 36, 288)-Net in Base 64 — Constructive

(15, 36, 288)-net in base 64, using

6 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- base change [i] based on digital (9, 36, 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]

(36−21, 36, 321)-Net in Base 64

(15, 36, 321)-net in base 64, using

16 times m-reduction [i] based on (15, 52, 321)-net in base 64, using
- base change [i] based on digital (2, 39, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(36−21, 36, 150748)-Net in Base 64 — Upper bound on s

There is no (15, 36, 150749)-net in base 64, because

1 times m-reduction [i] would yield (15, 35, 150749)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1645 587848 124662 384640 561660 703037 789195 753453 991315 079828 847446 > 64³⁵ [i]