MinT - Best Known (37−22, 37, s)-Nets in Base 64

Best Known (37−22, 37, s)-Nets in Base 64

(37−22, 37, 184)-Net over F₆₄ — Constructive and digital

Digital (15, 37, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. digital (3, 25, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(37−22, 37, 259)-Net over F₆₄ — Digital

Digital (15, 37, 259)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64³⁷, 259, F₆₄, 2, 22) (dual of [(259, 2), 481, 23]-NRT-code), using
- constructionÂ X applied to AG(2;F,489P) âŠ‚ AG(2;F,493P) [i] based on
  1. linear OOA(64³⁴, 256, F₆₄, 2, 22) (dual of [(256, 2), 478, 23]-NRT-code), using algebraic-geometric NRT-code AG(2;F,489P) [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 OOA(64³⁰, 256, F₆₄, 2, 18) (dual of [(256, 2), 482, 19]-NRT-code), using algebraic-geometric NRT-code AG(2;F,493P) [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257 (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]

(37−22, 37, 288)-Net in Base 64 — Constructive

(15, 37, 288)-net in base 64, using

5 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]

(37−22, 37, 321)-Net in Base 64

(15, 37, 321)-net in base 64, using

15 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]

(37−22, 37, 92679)-Net in Base 64 — Upper bound on s

There is no (15, 37, 92680)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 6 740673 927978 686146 059960 601655 413306 258654 850431 936923 247051 379776 > 64³⁷ [i]