MinT - Best Known (117−89, 117, s)-Nets in Base 16

Best Known (117−89, 117, s)-Nets in Base 16

(117−89, 117, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 117, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 117, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(117−89, 117, 66)-Net in Base 16 — Constructive

(28, 117, 66)-net in base 16, using

t-expansion [i] based on (25, 117, 66)-net in base 16, using
- net from sequence [i] based on (25, 65)-sequence in base 16, using
  - base expansion [i] based on digital (50, 65)-sequence over F₄, using
    - t-expansion [i] based on digital (49, 65)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
        T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(117−89, 117, 156)-Net over F₁₆ — Digital

Digital (28, 117, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 117, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(117−89, 117, 1694)-Net in Base 16 — Upper bound on s

There is no (28, 117, 1695)-net in base 16, because

1 times m-reduction [i] would yield (28, 116, 1695)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 47 799027 345167 243250 792747 678179 718027 354457 856614 537838 227412 240215 194993 770756 997617 005390 559695 686612 264460 827595 399579 340786 720896 569076 > 16¹¹⁶ [i]