MinT - Best Known (241−131, 241, s)-Nets in Base 4

Best Known (241−131, 241, s)-Nets in Base 4

(241−131, 241, 130)-Net over F₄ — Constructive and digital

Digital (110, 241, 130)-net over F₄, using

t-expansion [i] based on digital (105, 241, 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]

(241−131, 241, 165)-Net over F₄ — Digital

Digital (110, 241, 165)-net over F₄, using

t-expansion [i] based on digital (109, 241, 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]

(241−131, 241, 1342)-Net in Base 4 — Upper bound on s

There is no (110, 241, 1343)-net in base 4, because

1 times m-reduction [i] would yield (110, 240, 1343)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 250623 909691 327097 026367 811912 950528 075895 395350 937270 190198 462803 335673 432829 608684 078403 460653 349947 815267 456110 961817 688652 579314 749628 575640 > 4²⁴⁰ [i]