MinT - Best Known (13−8, 13, s)-Nets in Base 64

Best Known (13−8, 13, s)-Nets in Base 64

(13−8, 13, 145)-Net over F₆₄ — Constructive and digital

Digital (5, 13, 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, 9, 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]

(13−8, 13, 193)-Net over F₆₄ — Digital

Digital (5, 13, 193)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹³, 193, F₆₄, 8) (dual of [193, 180, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(64¹³, 195, F₆₄, 8) (dual of [195, 182, 9]-code), using
  - the BCH-code C(I) with length 195Â | 64²âˆ’1, defining interval IÂ =Â {1,17,33,…,113}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(13−8, 13, 258)-Net in Base 64 — Constructive

(5, 13, 258)-net in base 64, using

3 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]

(13−8, 13, 289)-Net in Base 64

(5, 13, 289)-net in base 64, using

3 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]

(13−8, 13, 26047)-Net in Base 64 — Upper bound on s

There is no (5, 13, 26048)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 302240 567102 323122 448561 > 64¹³ [i]