MinT - Best Known (19−11, 19, s)-Nets in Base 64

Best Known (19−11, 19, s)-Nets in Base 64

(19−11, 19, 195)-Net over F₆₄ — Constructive and digital

Digital (8, 19, 195)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 3, 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 (0, 5, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
3. digital (0, 11, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(19−11, 19, 260)-Net in Base 64 — Constructive

(8, 19, 260)-net in base 64, using

1 times m-reduction [i] based on (8, 20, 260)-net in base 64, using
- base change [i] based on digital (3, 15, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(19−11, 19, 266)-Net over F₆₄ — Digital

Digital (8, 19, 266)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹⁹, 266, F₆₄, 11) (dual of [266, 247, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(64¹⁹, 315, F₆₄, 11) (dual of [315, 296, 12]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 315Â | 64²âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(19−11, 19, 321)-Net in Base 64

(8, 19, 321)-net in base 64, using

5 times m-reduction [i] based on (8, 24, 321)-net in base 64, using
- base change [i] based on digital (2, 18, 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]

(19−11, 19, 131442)-Net in Base 64 — Upper bound on s

There is no (8, 19, 131443)-net in base 64, because

1 times m-reduction [i] would yield (8, 18, 131443)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 324 521269 273876 711157 527708 028728 > 64¹⁸ [i]