MinT - Best Known (242−137, 242, s)-Nets in Base 4

Best Known (242−137, 242, s)-Nets in Base 4

(242−137, 242, 130)-Net over F₄ — Constructive and digital

Digital (105, 242, 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]

(242−137, 242, 144)-Net over F₄ — Digital

Digital (105, 242, 144)-net over F₄, using

t-expansion [i] based on digital (91, 242, 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]

(242−137, 242, 1131)-Net in Base 4 — Upper bound on s

There is no (105, 242, 1132)-net in base 4, because

1 times m-reduction [i] would yield (105, 241, 1132)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 086827 906818 409691 248545 497029 500553 173271 915458 304960 915164 046657 864638 258733 762365 060720 533004 501865 241888 788592 707875 074953 468205 272533 747054 > 4²⁴¹ [i]