MinT - Best Known (256−145, 256, s)-Nets in Base 4

Best Known (256−145, 256, s)-Nets in Base 4

(256−145, 256, 130)-Net over F₄ — Constructive and digital

Digital (111, 256, 130)-net over F₄, using

t-expansion [i] based on digital (105, 256, 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]

(256−145, 256, 165)-Net over F₄ — Digital

Digital (111, 256, 165)-net over F₄, using

t-expansion [i] based on digital (109, 256, 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]

(256−145, 256, 1190)-Net in Base 4 — Upper bound on s

There is no (111, 256, 1191)-net in base 4, because

1 times m-reduction [i] would yield (111, 255, 1191)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3383 522404 951590 143305 682085 426103 862217 075057 336663 507043 853224 660976 159726 363224 008352 986896 639840 158932 047807 041022 595253 503254 802326 856322 829482 643032 > 4²⁵⁵ [i]