MinT - Best Known (51−37, 51, s)-Nets in Base 4

Best Known (51−37, 51, s)-Nets in Base 4

(51−37, 51, 30)-Net over F₄ — Constructive and digital

Digital (14, 51, 30)-net over F₄, using

t-expansion [i] based on digital (13, 51, 30)-net over F₄, using
- net from sequence [i] based on digital (13, 29)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
    - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(51−37, 51, 33)-Net over F₄ — Digital

Digital (14, 51, 33)-net over F₄, using

t-expansion [i] based on digital (13, 51, 33)-net over F₄, using
- net from sequence [i] based on digital (13, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
    - Drinfeld modules of rank 1 [i]

(51−37, 51, 81)-Net in Base 4 — Upper bound on s

There is no (14, 51, 82)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵¹, 82, S₄, 37), but
- the linear programming bound shows that MÂ â‰¥ 525829 445535 615750 659879 714919 334210 583084 997472 321206 747136 / 96139 990085 199610 191009 543175 > 4⁵¹ [i]