MinT - Best Known (59, 212, s)-Nets in Base 4

Best Known (59, 212, s)-Nets in Base 4

(59, 212, 66)-Net over F₄ — Constructive and digital

Digital (59, 212, 66)-net over F₄, using

t-expansion [i] based on digital (49, 212, 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]

(59, 212, 91)-Net over F₄ — Digital

Digital (59, 212, 91)-net over F₄, using

t-expansion [i] based on digital (50, 212, 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]

(59, 212, 338)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 212, 339)-net over F₄, because

1 times m-reduction [i] would yield digital (59, 211, 339)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹¹, 339, F₄, 152) (dual of [339, 128, 153]-code), but
  - residual code [i] would yield OA(4⁵⁹, 186, S₄, 38), but
    - the linear programming bound shows that MÂ â‰¥ 458116 895032 657940 162336 762311 305276 187123 578098 709475 576222 779546 009600 / 1 358774 703882 919180 099545 580638 520853 > 4⁵⁹ [i]

(59, 212, 395)-Net in Base 4 — Upper bound on s

There is no (59, 212, 396)-net in base 4, because

1 times m-reduction [i] would yield (59, 211, 396)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 190608 271934 785124 950816 950054 347326 247733 640200 354808 040094 916252 085420 051186 177928 260899 889729 231231 217141 898231 929323 604416 > 4²¹¹ [i]