MinT - Best Known (257−141, 257, s)-Nets in Base 4

Best Known (257−141, 257, s)-Nets in Base 4

(257−141, 257, 130)-Net over F₄ — Constructive and digital

Digital (116, 257, 130)-net over F₄, using

t-expansion [i] based on digital (105, 257, 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]

(257−141, 257, 168)-Net over F₄ — Digital

Digital (116, 257, 168)-net over F₄, using

t-expansion [i] based on digital (115, 257, 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]

(257−141, 257, 1369)-Net in Base 4 — Upper bound on s

There is no (116, 257, 1370)-net in base 4, because

1 times m-reduction [i] would yield (116, 256, 1370)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13453 971325 557882 751153 920144 429256 367509 424173 604305 863199 314712 175834 862910 385939 651280 932915 456746 004327 975726 635791 882304 215972 767273 520190 252959 627484 > 4²⁵⁶ [i]