MinT - Best Known (254−137, 254, s)-Nets in Base 4

Best Known (254−137, 254, s)-Nets in Base 4

(254−137, 254, 130)-Net over F₄ — Constructive and digital

Digital (117, 254, 130)-net over F₄, using

t-expansion [i] based on digital (105, 254, 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]

(254−137, 254, 168)-Net over F₄ — Digital

Digital (117, 254, 168)-net over F₄, using

t-expansion [i] based on digital (115, 254, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(254−137, 254, 1459)-Net in Base 4 — Upper bound on s

There is no (117, 254, 1460)-net in base 4, because

1 times m-reduction [i] would yield (117, 253, 1460)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 211 538945 741841 252083 280758 920771 956056 515476 247793 135451 048682 902662 785698 605326 466114 924332 802205 295750 566520 139487 247895 627925 538494 278714 094623 919400 > 4²⁵³ [i]