MinT - Best Known (71−66, 71, s)-Nets in Base 16

Best Known (71−66, 71, s)-Nets in Base 16

(71−66, 71, 49)-Net over F₁₆ — Constructive and digital

Digital (5, 71, 49)-net over F₁₆, using

net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
  - a function field by SÃ©mirat [i]

(71−66, 71, 158)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (5, 71, 159)-net over F₁₆, because

2 times m-reduction [i] would yield digital (5, 69, 159)-net over F₁₆, but
- extracting embedded orthogonal array [i] would yield linear OA(16⁶⁹, 159, F₁₆, 64) (dual of [159, 90, 65]-code), but
  - residual code [i] would yield OA(16⁵, 94, S₁₆, 4), but
    - the linear programming bound shows that MÂ â‰¥ 2179 760128 / 2071 > 16⁵ [i]

(71−66, 71, 183)-Net in Base 16 — Upper bound on s

There is no (5, 71, 184)-net in base 16, because

extracting embedded orthogonal array [i] would yield OA(16⁷¹, 184, S₁₆, 66), but
- the linear programming bound shows that MÂ â‰¥ 736 115296 179792 197406 603832 460392 758683 771472 138505 152276 194264 299873 685131 262688 641541 821786 805799 322346 112582 861954 482176 / 22 908652 434206 366957 779935 808052 259965 > 16⁷¹ [i]