MinT - Best Known (13, 13+43, s)-Nets in Base 64

Best Known (13, 13+43, s)-Nets in Base 64

(13, 13+43, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 56, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 56, 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]

(13, 13+43, 216)-Net in Base 64 — Constructive

(13, 56, 216)-net in base 64, using

base change [i] based on digital (5, 48, 216)-net over F₁₂₈, using
- net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
    - a function field by SÃ©mirat [i]

(13, 13+43, 257)-Net over F₆₄ — Digital

Digital (13, 56, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 56, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(13, 13+43, 7396)-Net in Base 64 — Upper bound on s

There is no (13, 56, 7397)-net in base 64, because

1 times m-reduction [i] would yield (13, 55, 7397)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2191 309272 841966 299910 464565 803643 228581 444500 451107 909028 208666 800630 674948 823140 073884 261233 294432 > 64⁵⁵ [i]