MinT - Best Known (2, 2+36, s)-Nets in Base 16

Best Known (2, 2+36, s)-Nets in Base 16

(2, 2+36, 33)-Net over F₁₆ — Constructive and digital

Digital (2, 38, 33)-net over F₁₆, using

net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
  - a function field by SÃ©mirat [i]

(2, 2+36, 50)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (2, 38, 51)-net over F₁₆, because

4 times m-reduction [i] would yield digital (2, 34, 51)-net over F₁₆, but
- extracting embedded orthogonal array [i] would yield linear OA(16³⁴, 51, F₁₆, 32) (dual of [51, 17, 33]-code), but
  - residual code [i] would yield OA(16², 18, S₁₆, 2), but
    - bound for OAs with strength kÂ =Â 2 [i]
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 271 > 16² [i]

(2, 2+36, 94)-Net in Base 16 — Upper bound on s

There is no (2, 38, 95)-net in base 16, because

extracting embedded orthogonal array [i] would yield OA(16³⁸, 95, S₁₆, 36), but
- the linear programming bound shows that MÂ â‰¥ 3 265990 803455 640149 447634 567433 080672 094117 566036 219755 757568 / 571 680028 473133 > 16³⁸ [i]