MinT - Best Known (257−151, 257, s)-Nets in Base 4

Best Known (257−151, 257, s)-Nets in Base 4

(257−151, 257, 130)-Net over F₄ — Constructive and digital

Digital (106, 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]

(257−151, 257, 144)-Net over F₄ — Digital

Digital (106, 257, 144)-net over F₄, using

t-expansion [i] based on digital (91, 257, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(257−151, 257, 1026)-Net in Base 4 — Upper bound on s

There is no (106, 257, 1027)-net in base 4, because

1 times m-reduction [i] would yield (106, 256, 1027)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13408 792342 351822 745118 823784 939474 560080 420485 547186 447398 100395 578114 166176 363780 320288 503102 844719 732772 790301 273289 287574 675528 812095 515961 720778 003968 > 4²⁵⁶ [i]