MinT - Best Known (33−31, 33, s)-Nets in Base 16

Best Known (33−31, 33, s)-Nets in Base 16

(33−31, 33, 33)-Net over F₁₆ — Constructive and digital

Digital (2, 33, 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]

(33−31, 33, 65)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (2, 33, 66)-net over F₁₆, because

1 times m-reduction [i] would yield digital (2, 32, 66)-net over F₁₆, but
- extracting embedded orthogonal array [i] would yield linear OA(16³², 66, F₁₆, 30) (dual of [66, 34, 31]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(16³¹, 34, F₁₆, 30) (dual of [34, 3, 31]-code), but
      - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
    2. linear OA(16³⁴, 66, F₁₆, 32) (dual of [66, 32, 33]-code), but
      - discarding factors / shortening the dual code 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]

(33−31, 33, 112)-Net in Base 16 — Upper bound on s

There is no (2, 33, 113)-net in base 16, because

1 times m-reduction [i] would yield (2, 32, 113)-net in base 16, but
- extracting embedded orthogonal array [i] would yield OA(16³², 113, S₁₆, 30), but
  - the linear programming bound shows that MÂ â‰¥ 717 553682 135575 190882 837481 077483 153553 381648 236544 / 2 087375 178989 > 16³² [i]