MinT - Best Known (255−133, 255, s)-Nets in Base 4

Best Known (255−133, 255, s)-Nets in Base 4

(255−133, 255, 130)-Net over F₄ — Constructive and digital

Digital (122, 255, 130)-net over F₄, using

t-expansion [i] based on digital (105, 255, 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]

(255−133, 255, 168)-Net over F₄ — Digital

Digital (122, 255, 168)-net over F₄, using

t-expansion [i] based on digital (115, 255, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(255−133, 255, 1704)-Net in Base 4 — Upper bound on s

There is no (122, 255, 1705)-net in base 4, because

1 times m-reduction [i] would yield (122, 254, 1705)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 869 924361 362651 816153 050377 461681 524318 543710 412126 408269 436728 434826 549949 048437 685515 876784 038260 361996 229094 454621 357339 745033 932757 788547 971093 914200 > 4²⁵⁴ [i]