MinT - Best Known (56, 56+160, s)-Nets in Base 4

Best Known (56, 56+160, s)-Nets in Base 4

(56, 56+160, 66)-Net over F₄ — Constructive and digital

Digital (56, 216, 66)-net over F₄, using

t-expansion [i] based on digital (49, 216, 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]

(56, 56+160, 91)-Net over F₄ — Digital

Digital (56, 216, 91)-net over F₄, using

t-expansion [i] based on digital (50, 216, 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]

(56, 56+160, 258)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 216, 259)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²¹⁶, 259, F₄, 160) (dual of [259, 43, 161]-code), but
- residual code [i] would yield OA(4⁵⁶, 98, S₄, 40), but
  - the linear programming bound shows that MÂ â‰¥ 108 858270 389765 832134 899494 245555 240896 445146 215984 684712 894959 792949 029119 197184 / 19965 002432 218887 804054 871233 298540 608350 321175 > 4⁵⁶ [i]

(56, 56+160, 368)-Net in Base 4 — Upper bound on s

There is no (56, 216, 369)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13178 776832 113111 000447 060807 137137 275164 892668 387271 868798 321959 172743 408663 718779 181846 032565 711202 003137 940171 066346 530254 937013 > 4²¹⁶ [i]