MinT - Best Known (138−90, 138, s)-Nets in Base 4

Best Known (138−90, 138, s)-Nets in Base 4

(138−90, 138, 56)-Net over F₄ — Constructive and digital

Digital (48, 138, 56)-net over F₄, using

t-expansion [i] based on digital (33, 138, 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]

(138−90, 138, 81)-Net over F₄ — Digital

Digital (48, 138, 81)-net over F₄, using

t-expansion [i] based on digital (46, 138, 81)-net over F₄, using
- net from sequence [i] based on digital (46, 80)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(138−90, 138, 376)-Net in Base 4 — Upper bound on s

There is no (48, 138, 377)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 125979 146486 305131 181993 632263 983598 727942 740292 042791 880183 813871 612942 389454 830304 > 4¹³⁸ [i]