MinT - Best Known (238−178, 238, s)-Nets in Base 4

Best Known (238−178, 238, s)-Nets in Base 4

(238−178, 238, 66)-Net over F₄ — Constructive and digital

Digital (60, 238, 66)-net over F₄, using

t-expansion [i] based on digital (49, 238, 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]

(238−178, 238, 91)-Net over F₄ — Digital

Digital (60, 238, 91)-net over F₄, using

t-expansion [i] based on digital (50, 238, 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]

(238−178, 238, 257)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 238, 258)-net over F₄, because

2 times m-reduction [i] would yield digital (60, 236, 258)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁶, 258, F₄, 176) (dual of [258, 22, 177]-code), but
  - residual code [i] would yield OA(4⁶⁰, 81, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 278 016363 629983 093645 810481 419845 575227 299067 854848 / 143 941903 015625 > 4⁶⁰ [i]

(238−178, 238, 391)-Net in Base 4 — Upper bound on s

There is no (60, 238, 392)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 228745 278567 798420 115177 571222 995080 695181 718971 254617 831603 455951 201292 115358 604313 440379 981485 583243 019144 410841 870482 475255 806549 042340 808240 > 4²³⁸ [i]