MinT - Best Known (129−101, 129, s)-Nets in Base 16

Best Known (129−101, 129, s)-Nets in Base 16

(129−101, 129, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 129, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 129, 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]

(129−101, 129, 66)-Net in Base 16 — Constructive

(28, 129, 66)-net in base 16, using

t-expansion [i] based on (25, 129, 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]

(129−101, 129, 156)-Net over F₁₆ — Digital

Digital (28, 129, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 129, 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]

(129−101, 129, 1543)-Net in Base 16 — Upper bound on s

There is no (28, 129, 1544)-net in base 16, because

1 times m-reduction [i] would yield (28, 128, 1544)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 13808 309916 794906 330372 947223 751788 571736 095786 697664 616225 434116 385998 696955 803238 752170 722493 440659 589234 510380 510834 000517 674588 208864 124535 511502 417376 > 16¹²⁸ [i]