MinT - Best Known (256−149, 256, s)-Nets in Base 4

Best Known (256−149, 256, s)-Nets in Base 4

(256−149, 256, 130)-Net over F₄ — Constructive and digital

Digital (107, 256, 130)-net over F₄, using

t-expansion [i] based on digital (105, 256, 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]

(256−149, 256, 144)-Net over F₄ — Digital

Digital (107, 256, 144)-net over F₄, using

t-expansion [i] based on digital (91, 256, 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]

(256−149, 256, 1063)-Net in Base 4 — Upper bound on s

There is no (107, 256, 1064)-net in base 4, because

1 times m-reduction [i] would yield (107, 255, 1064)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3484 798391 556205 210536 994918 740581 986679 851027 043311 827128 795186 493542 490961 688022 207145 910847 569240 059892 648051 935434 183450 507895 561204 560264 967831 072192 > 4²⁵⁵ [i]