MinT - Best Known (9, 21, s)-Nets in Base 64

Best Known (9, 21, s)-Nets in Base 64

(9, 21, 177)-Net over F₆₄ — Constructive and digital

Digital (9, 21, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 21, 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]

(9, 21, 260)-Net in Base 64 — Constructive

(9, 21, 260)-net in base 64, using

3 times m-reduction [i] based on (9, 24, 260)-net in base 64, using
- base change [i] based on digital (3, 18, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(9, 21, 291)-Net over F₆₄ — Digital

Digital (9, 21, 291)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²¹, 291, F₆₄, 12) (dual of [291, 270, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(64²¹, 315, F₆₄, 12) (dual of [315, 294, 13]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 315Â | 64²âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(9, 21, 321)-Net in Base 64

(9, 21, 321)-net in base 64, using

7 times m-reduction [i] based on (9, 28, 321)-net in base 64, using
- base change [i] based on digital (2, 21, 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]

(9, 21, 99655)-Net in Base 64 — Upper bound on s

There is no (9, 21, 99656)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 85 074492 468930 506965 847933 249663 325943 > 64²¹ [i]