MinT - Best Known (123−88, 123, s)-Nets in Base 4

Best Known (123−88, 123, s)-Nets in Base 4

(123−88, 123, 56)-Net over F₄ — Constructive and digital

Digital (35, 123, 56)-net over F₄, using

t-expansion [i] based on digital (33, 123, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(123−88, 123, 65)-Net over F₄ — Digital

Digital (35, 123, 65)-net over F₄, using

t-expansion [i] based on digital (33, 123, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(123−88, 123, 149)-Net in Base 4 — Upper bound on s

There is no (35, 123, 150)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹²³, 150, S₄, 88), but
- the linear programming bound shows that MÂ â‰¥ 22915 666113 439092 609099 444232 414491 754857 959622 104295 992957 494311 648680 329878 580928 609806 974976 / 136 168406 561048 662801 > 4¹²³ [i]