MinT - Best Known (82−59, 82, s)-Nets in Base 64

Best Known (82−59, 82, s)-Nets in Base 64

(82−59, 82, 177)-Net over F₆₄ — Constructive and digital

Digital (23, 82, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 82, 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]

(82−59, 82, 288)-Net in Base 64 — Constructive

(23, 82, 288)-net in base 64, using

t-expansion [i] based on (22, 82, 288)-net in base 64, using
- 9 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
  - base change [i] based on digital (9, 78, 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]

(82−59, 82, 342)-Net over F₆₄ — Digital

Digital (23, 82, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 82, 342)-net over F₆₄, using
- net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
    - a curve from the manYPoints database [i]

(82−59, 82, 20526)-Net in Base 64 — Upper bound on s

There is no (23, 82, 20527)-net in base 64, because

1 times m-reduction [i] would yield (23, 81, 20527)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 199 822323 949380 208910 086064 342118 485042 869349 153490 448118 769679 994920 939935 294692 135384 538581 242234 689096 366114 233968 975637 403400 194062 464700 977108 > 64⁸¹ [i]