MinT - Best Known (82−71, 82, s)-Nets in Base 4

Best Known (82−71, 82, s)-Nets in Base 4

(82−71, 82, 27)-Net over F₄ — Constructive and digital

Digital (11, 82, 27)-net over F₄, using

t-expansion [i] based on digital (10, 82, 27)-net over F₄, using
- net from sequence [i] based on digital (10, 26)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
    - an explicitly constructive algebraic function field [i]

(82−71, 82, 50)-Net over F₄ — Upper bound on s (digital)

There is no digital (11, 82, 51)-net over F₄, because

35 times m-reduction [i] would yield digital (11, 47, 51)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁴⁷, 51, F₄, 36) (dual of [51, 4, 37]-code), but
  - a result by Landjev, Maruta, and Hill [i]

(82−71, 82, 53)-Net in Base 4 — Upper bound on s

There is no (11, 82, 54)-net in base 4, because

34 times m-reduction [i] would yield (11, 48, 54)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁴⁸, 54, S₄, 37), but
  - the linear programming bound shows that MÂ â‰¥ 5 070602 400912 917605 986812 821504 / 55 > 4⁴⁸ [i]