MinT - Best Known (240−127, 240, s)-Nets in Base 4

Best Known (240−127, 240, s)-Nets in Base 4

(240−127, 240, 130)-Net over F₄ — Constructive and digital

Digital (113, 240, 130)-net over F₄, using

t-expansion [i] based on digital (105, 240, 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]

(240−127, 240, 165)-Net over F₄ — Digital

Digital (113, 240, 165)-net over F₄, using

t-expansion [i] based on digital (109, 240, 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]

(240−127, 240, 1506)-Net in Base 4 — Upper bound on s

There is no (113, 240, 1507)-net in base 4, because

1 times m-reduction [i] would yield (113, 239, 1507)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 793000 746415 784118 315461 597820 629706 444668 172043 616245 417458 496489 582525 745757 610367 471493 098132 734194 949441 523839 484486 618275 715314 676719 087360 > 4²³⁹ [i]