MinT - Best Known (245−129, 245, s)-Nets in Base 4

Best Known (245−129, 245, s)-Nets in Base 4

(245−129, 245, 130)-Net over F₄ — Constructive and digital

Digital (116, 245, 130)-net over F₄, using

t-expansion [i] based on digital (105, 245, 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]

(245−129, 245, 168)-Net over F₄ — Digital

Digital (116, 245, 168)-net over F₄, using

t-expansion [i] based on digital (115, 245, 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]

(245−129, 245, 1571)-Net in Base 4 — Upper bound on s

There is no (116, 245, 1572)-net in base 4, because

1 times m-reduction [i] would yield (116, 244, 1572)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 818 588862 763792 112440 700876 814499 465325 640497 401008 940747 375995 467642 179056 504394 573742 488209 086985 479913 271186 098953 072642 928888 594628 300625 539192 > 4²⁴⁴ [i]