MinT - Best Known (14−9, 14, s)-Nets in Base 64

Best Known (14−9, 14, s)-Nets in Base 64

(14−9, 14, 145)-Net over F₆₄ — Constructive and digital

Digital (5, 14, 145)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 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 (1, 10, 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]

(14−9, 14, 146)-Net over F₆₄ — Digital

Digital (5, 14, 146)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64¹⁴, 146, F₆₄, 2, 9) (dual of [(146, 2), 278, 10]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(64⁴, 65, F₆₄, 2, 4) (dual of [(65, 2), 126, 5]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(2;126,64) [i]
  2. linear OOA(64¹⁰, 81, F₆₄, 2, 9) (dual of [(81, 2), 152, 10]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,152P) [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(14−9, 14, 258)-Net in Base 64 — Constructive

(5, 14, 258)-net in base 64, using

2 times m-reduction [i] based on (5, 16, 258)-net in base 64, using
- base change [i] based on digital (1, 12, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(14−9, 14, 289)-Net in Base 64

(5, 14, 289)-net in base 64, using

2 times m-reduction [i] based on (5, 16, 289)-net in base 64, using
- base change [i] based on digital (1, 12, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(14−9, 14, 26047)-Net in Base 64 — Upper bound on s

There is no (5, 14, 26048)-net in base 64, because

1 times m-reduction [i] would yield (5, 13, 26048)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 302240 567102 323122 448561 > 64¹³ [i]