MinT - Best Known (58, 58+170, s)-Nets in Base 4

Best Known (58, 58+170, s)-Nets in Base 4

(58, 58+170, 66)-Net over F₄ — Constructive and digital

Digital (58, 228, 66)-net over F₄, using

t-expansion [i] based on digital (49, 228, 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]

(58, 58+170, 91)-Net over F₄ — Digital

Digital (58, 228, 91)-net over F₄, using

t-expansion [i] based on digital (50, 228, 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]

(58, 58+170, 256)-Net over F₄ — Upper bound on s (digital)

There is no digital (58, 228, 257)-net over F₄, because

2 times m-reduction [i] would yield digital (58, 226, 257)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁶, 257, F₄, 168) (dual of [257, 31, 169]-code), but
  - residual code [i] would yield OA(4⁵⁸, 88, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 158 237175 107336 529531 271038 987459 138072 539269 830656 314178 535424 / 1500 577564 280935 407980 323611 > 4⁵⁸ [i]

(58, 58+170, 379)-Net in Base 4 — Upper bound on s

There is no (58, 228, 380)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 215809 157726 486382 049034 808841 307645 770206 590318 985855 555183 732866 567389 018724 334211 063046 816177 902269 846159 342942 093909 055758 785419 869574 > 4²²⁸ [i]