MinT - Best Known (3, 3+51, s)-Nets in Base 16

Best Known (3, 3+51, s)-Nets in Base 16

(3, 3+51, 38)-Net over F₁₆ — Constructive and digital

Digital (3, 54, 38)-net over F₁₆, using

net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
  - a function field by SÃ©mirat [i]

(3, 3+51, 67)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (3, 54, 68)-net over F₁₆, because

3 times m-reduction [i] would yield digital (3, 51, 68)-net over F₁₆, but
- extracting embedded orthogonal array [i] would yield linear OA(16⁵¹, 68, F₁₆, 48) (dual of [68, 17, 49]-code), but
  - residual code [i] would yield OA(16³, 19, S₁₆, 3), but
    - 1 times truncation [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]

(3, 3+51, 109)-Net in Base 16 — Upper bound on s

There is no (3, 54, 110)-net in base 16, because

1 times m-reduction [i] would yield (3, 53, 110)-net in base 16, but
- extracting embedded orthogonal array [i] would yield OA(16⁵³, 110, S₁₆, 50), but
  - the linear programming bound shows that MÂ â‰¥ 157604 905779 606138 415583 163697 716541 301437 424228 323859 211245 536998 164846 300572 418048 / 23 406013 355287 257923 > 16⁵³ [i]