MinT - Best Known (257−131, 257, s)-Nets in Base 4

Best Known (257−131, 257, s)-Nets in Base 4

(257−131, 257, 130)-Net over F₄ — Constructive and digital

Digital (126, 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]

(257−131, 257, 176)-Net over F₄ — Digital

Digital (126, 257, 176)-net over F₄, using

t-expansion [i] based on digital (125, 257, 176)-net over F₄, using
- net from sequence [i] based on digital (125, 175)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 125 and N(F) ≥ 176, using
    - Drinfeld modules of rank 1 [i]

(257−131, 257, 1909)-Net in Base 4 — Upper bound on s

There is no (126, 257, 1910)-net in base 4, because

1 times m-reduction [i] would yield (126, 256, 1910)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13740 870083 654298 734518 481327 230119 995593 480463 292306 427830 591539 968545 409979 637837 972203 121139 894591 232808 609867 225069 852473 376075 453969 490540 222838 993125 > 4²⁵⁶ [i]