MinT - Best Known (123−95, 123, s)-Nets in Base 16

Best Known (123−95, 123, s)-Nets in Base 16

(123−95, 123, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 123, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 123, 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]

(123−95, 123, 66)-Net in Base 16 — Constructive

(28, 123, 66)-net in base 16, using

t-expansion [i] based on (25, 123, 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]

(123−95, 123, 156)-Net over F₁₆ — Digital

Digital (28, 123, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 123, 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]

(123−95, 123, 1609)-Net in Base 16 — Upper bound on s

There is no (28, 123, 1610)-net in base 16, because

1 times m-reduction [i] would yield (28, 122, 1610)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 816 241009 307554 130812 203430 489033 369344 332991 730052 073846 159120 901897 904489 844699 688993 409380 269676 804636 489426 477073 544791 909414 224928 231355 686176 > 16¹²² [i]