MinT - Best Known (251−191, 251, s)-Nets in Base 4

Best Known (251−191, 251, s)-Nets in Base 4

(251−191, 251, 66)-Net over F₄ — Constructive and digital

Digital (60, 251, 66)-net over F₄, using

t-expansion [i] based on digital (49, 251, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(251−191, 251, 91)-Net over F₄ — Digital

Digital (60, 251, 91)-net over F₄, using

t-expansion [i] based on digital (50, 251, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(251−191, 251, 250)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 251, 251)-net over F₄, because

7 times m-reduction [i] would yield digital (60, 244, 251)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁴, 251, F₄, 184) (dual of [251, 7, 185]-code), but
  - residual code [i] would yield OA(4⁶⁰, 66, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 340 282366 920938 463463 374607 431768 211456 / 235 > 4⁶⁰ [i]

(251−191, 251, 388)-Net in Base 4 — Upper bound on s

There is no (60, 251, 389)-net in base 4, because

1 times m-reduction [i] would yield (60, 250, 389)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 474189 846756 567888 972205 804198 821313 579148 994235 297105 713798 026966 308063 673806 517254 554365 433289 636514 041563 159909 091459 177387 998633 616646 442056 911744 > 4²⁵⁰ [i]