MinT - Best Known (223−167, 223, s)-Nets in Base 4

Best Known (223−167, 223, s)-Nets in Base 4

(223−167, 223, 66)-Net over F₄ — Constructive and digital

Digital (56, 223, 66)-net over F₄, using

t-expansion [i] based on digital (49, 223, 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]

(223−167, 223, 91)-Net over F₄ — Digital

Digital (56, 223, 91)-net over F₄, using

t-expansion [i] based on digital (50, 223, 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]

(223−167, 223, 241)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 223, 242)-net over F₄, because

3 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]

(223−167, 223, 366)-Net in Base 4 — Upper bound on s

There is no (56, 223, 367)-net in base 4, because

1 times m-reduction [i] would yield (56, 222, 367)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54 301759 077598 690145 321851 980028 297538 390982 121259 397025 621384 131293 215238 404124 203453 903290 201055 682787 376636 326715 363758 616574 028304 > 4²²² [i]