MinT - Best Known (29, 126, s)-Nets in Base 16

Best Known (29, 126, s)-Nets in Base 16

(29, 126, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 126, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 126, 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]

(29, 126, 66)-Net in Base 16 — Constructive

(29, 126, 66)-net in base 16, using

t-expansion [i] based on (25, 126, 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]

(29, 126, 161)-Net over F₁₆ — Digital

Digital (29, 126, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(29, 126, 1681)-Net in Base 16 — Upper bound on s

There is no (29, 126, 1682)-net in base 16, because

1 times m-reduction [i] would yield (29, 125, 1682)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 351033 416442 887790 169068 984566 336177 080454 157160 206432 640460 738413 363129 144285 895083 971309 039559 331632 795131 072591 316292 722093 364000 086259 495996 355316 > 16¹²⁵ [i]