MinT - Best Known (60, 60+185, s)-Nets in Base 4

Best Known (60, 60+185, s)-Nets in Base 4

(60, 60+185, 66)-Net over F₄ — Constructive and digital

Digital (60, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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]

(60, 60+185, 91)-Net over F₄ — Digital

Digital (60, 245, 91)-net over F₄, using

t-expansion [i] based on digital (50, 245, 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]

(60, 60+185, 250)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 245, 251)-net over F₄, because

1 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]

(60, 60+185, 389)-Net in Base 4 — Upper bound on s

There is no (60, 245, 390)-net in base 4, because

1 times m-reduction [i] would yield (60, 244, 390)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 843 189339 386963 070819 923299 378280 586494 891167 799035 979950 625743 812030 719161 524216 948582 945049 507048 936112 521075 597436 806799 843821 560382 950453 118960 > 4²⁴⁴ [i]