MinT - Best Known (253−193, 253, s)-Nets in Base 4

Best Known (253−193, 253, s)-Nets in Base 4

(253−193, 253, 66)-Net over F₄ — Constructive and digital

Digital (60, 253, 66)-net over F₄, using

t-expansion [i] based on digital (49, 253, 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]

(253−193, 253, 91)-Net over F₄ — Digital

Digital (60, 253, 91)-net over F₄, using

t-expansion [i] based on digital (50, 253, 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]

(253−193, 253, 250)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 253, 251)-net over F₄, because

9 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]

(253−193, 253, 255)-Net in Base 4 — Upper bound on s

There is no (60, 253, 256)-net in base 4, because

1 times m-reduction [i] would yield (60, 252, 256)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²⁵², 256, S₄, 192), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 13407 807929 942597 099574 024998 205846 127479 365820 592393 377723 561443 721764 030073 546976 801874 298166 903427 690031 858186 486050 853753 882811 946569 946433 649006 084096 / 193 > 4²⁵² [i]