MinT - Best Known (56, 56+165, s)-Nets in Base 4

Best Known (56, 56+165, s)-Nets in Base 4

(56, 56+165, 66)-Net over F₄ — Constructive and digital

Digital (56, 221, 66)-net over F₄, using

t-expansion [i] based on digital (49, 221, 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]

(56, 56+165, 91)-Net over F₄ — Digital

Digital (56, 221, 91)-net over F₄, using

t-expansion [i] based on digital (50, 221, 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]

(56, 56+165, 241)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 221, 242)-net over F₄, because

1 times m-reduction [i] would yield digital (56, 220, 242)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁰, 242, F₄, 164) (dual of [242, 22, 165]-code), but
  - residual code [i] would yield OA(4⁵⁶, 77, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 2852 409624 643877 869804 773184 242737 249513 373696 / 530979 549763 > 4⁵⁶ [i]

(56, 56+165, 366)-Net in Base 4 — Upper bound on s

There is no (56, 221, 367)-net in base 4, because

1 times m-reduction [i] would yield (56, 220, 367)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 016883 943997 794394 171840 577083 462264 423847 329485 990918 241330 933689 295259 866991 937259 608940 270328 860484 374514 032221 474566 593376 284450 > 4²²⁰ [i]