MinT - Best Known (239−127, 239, s)-Nets in Base 4

Best Known (239−127, 239, s)-Nets in Base 4

(239−127, 239, 130)-Net over F₄ — Constructive and digital

Digital (112, 239, 130)-net over F₄, using

t-expansion [i] based on digital (105, 239, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(239−127, 239, 165)-Net over F₄ — Digital

Digital (112, 239, 165)-net over F₄, using

t-expansion [i] based on digital (109, 239, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(239−127, 239, 1472)-Net in Base 4 — Upper bound on s

There is no (112, 239, 1473)-net in base 4, because

1 times m-reduction [i] would yield (112, 238, 1473)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 197450 069615 656819 441136 525640 500954 970113 162339 297334 898845 639272 768491 453623 427871 163961 734314 651152 841387 273840 256233 920101 090778 066317 553600 > 4²³⁸ [i]