MinT - Best Known (230−174, 230, s)-Nets in Base 4

Best Known (230−174, 230, s)-Nets in Base 4

(230−174, 230, 66)-Net over F₄ — Constructive and digital

Digital (56, 230, 66)-net over F₄, using

t-expansion [i] based on digital (49, 230, 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]

(230−174, 230, 91)-Net over F₄ — Digital

Digital (56, 230, 91)-net over F₄, using

t-expansion [i] based on digital (50, 230, 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]

(230−174, 230, 234)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 230, 235)-net over F₄, because

2 times m-reduction [i] would yield digital (56, 228, 235)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁸, 235, F₄, 172) (dual of [235, 7, 173]-code), but
  - residual code [i] would yield OA(4⁵⁶, 62, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 26 916866 914644 546426 302092 970676 977664 / 4675 > 4⁵⁶ [i]

(230−174, 230, 364)-Net in Base 4 — Upper bound on s

There is no (56, 230, 365)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 483383 716224 262997 540259 077868 778517 761459 668096 338499 259233 655812 861389 498018 267046 387611 375146 402797 733763 116638 313795 811659 915485 239728 > 4²³⁰ [i]