MinT - Best Known (254−131, 254, s)-Nets in Base 4

Best Known (254−131, 254, s)-Nets in Base 4

(254−131, 254, 130)-Net over F₄ — Constructive and digital

Digital (123, 254, 130)-net over F₄, using

t-expansion [i] based on digital (105, 254, 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]

(254−131, 254, 168)-Net over F₄ — Digital

Digital (123, 254, 168)-net over F₄, using

t-expansion [i] based on digital (115, 254, 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]

(254−131, 254, 1787)-Net in Base 4 — Upper bound on s

There is no (123, 254, 1788)-net in base 4, because

1 times m-reduction [i] would yield (123, 253, 1788)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 211 956694 391453 424050 327056 834242 513582 941985 433924 051850 503119 289105 126516 591310 043846 722012 893815 090430 181525 535883 281725 293366 269914 395413 213806 582305 > 4²⁵³ [i]