MinT - Best Known (51−45, 51, s)-Nets in Base 4

Best Known (51−45, 51, s)-Nets in Base 4

(51−45, 51, 17)-Net over F₄ — Constructive and digital

Digital (6, 51, 17)-net over F₄, using

t-expansion [i] based on digital (5, 51, 17)-net over F₄, using
- net from sequence [i] based on digital (5, 16)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(51−45, 51, 20)-Net over F₄ — Digital

Digital (6, 51, 20)-net over F₄, using

net from sequence [i] based on digital (6, 19)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 6 and N(F) ≥ 20, using
  - Drinfeld modules of rank 1 [i]

(51−45, 51, 30)-Net over F₄ — Upper bound on s (digital)

There is no digital (6, 51, 31)-net over F₄, because

25 times m-reduction [i] would yield digital (6, 26, 31)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁶, 31, F₄, 20) (dual of [31, 5, 21]-code), but
  - â€œBouâ€ bound on codes from Brouwerâ€™s database [i]

(51−45, 51, 33)-Net in Base 4 — Upper bound on s

There is no (6, 51, 34)-net in base 4, because

22 times m-reduction [i] would yield (6, 29, 34)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²⁹, 34, S₄, 23), but
  - the linear programming bound shows that MÂ â‰¥ 129 127208 515966 861312 / 435 > 4²⁹ [i]