MinT - Best Known (122−93, 122, s)-Nets in Base 16

Best Known (122−93, 122, s)-Nets in Base 16

(122−93, 122, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 122, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 122, 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]

(122−93, 122, 66)-Net in Base 16 — Constructive

(29, 122, 66)-net in base 16, using

t-expansion [i] based on (25, 122, 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]

(122−93, 122, 161)-Net over F₁₆ — Digital

Digital (29, 122, 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]

(122−93, 122, 1738)-Net in Base 16 — Upper bound on s

There is no (29, 122, 1739)-net in base 16, because

1 times m-reduction [i] would yield (29, 121, 1739)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 50 576196 912491 414677 029621 055778 795375 885401 303742 378519 131297 303626 322073 001370 674572 732689 589552 074074 691328 160623 625441 884841 390373 203618 695536 > 16¹²¹ [i]