MinT - Best Known (12−10, 12, s)-Nets in Base 4

Best Known (12−10, 12, s)-Nets in Base 4

(12−10, 12, 10)-Net over F₄ — Constructive and digital

Digital (2, 12, 10)-net over F₄, using

net from sequence [i] based on digital (2, 9)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
  - an explicitly constructive algebraic function field [i]

(12−10, 12, 14)-Net over F₄ — Upper bound on s (digital)

There is no digital (2, 12, 15)-net over F₄, because

2 times m-reduction [i] would yield digital (2, 10, 15)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁰, 15, F₄, 8) (dual of [15, 5, 9]-code), but
  - residual code [i] would yield OA(4², 6, S₄, 2), but
    - bound for OAs with strength kÂ =Â 2 [i]
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 19 > 4² [i]

(12−10, 12, 17)-Net in Base 4 — Upper bound on s

There is no (2, 12, 18)-net in base 4, because

extracting embedded OOA [i] would yield OOA(4¹², 18, S₄, 2, 10), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 168853 625785 810944 / 9612 299375 > 4¹² [i]