MinT - Best Known (72−58, 72, s)-Nets in Base 64

Best Known (72−58, 72, s)-Nets in Base 64

(72−58, 72, 177)-Net over F₆₄ — Constructive and digital

Digital (14, 72, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 72, 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]

(72−58, 72, 192)-Net in Base 64 — Constructive

(14, 72, 192)-net in base 64, using

5 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
- base change [i] based on digital (3, 66, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(72−58, 72, 257)-Net over F₆₄ — Digital

Digital (14, 72, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 72, 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]

(72−58, 72, 5636)-Net in Base 64 — Upper bound on s

There is no (14, 72, 5637)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 11142 445719 324421 119838 310088 132211 941678 400381 332346 796563 356246 588283 088274 488693 603669 711043 793815 967807 735858 144942 607455 684096 > 64⁷² [i]