MinT - Best Known (43−32, 43, s)-Nets in Base 4

Best Known (43−32, 43, s)-Nets in Base 4

(43−32, 43, 27)-Net over F₄ — Constructive and digital

Digital (11, 43, 27)-net over F₄, using

t-expansion [i] based on digital (10, 43, 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]

(43−32, 43, 54)-Net over F₄ — Upper bound on s (digital)

There is no digital (11, 43, 55)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4⁴³, 55, F₄, 32) (dual of [55, 12, 33]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4⁴², 47, F₄, 32) (dual of [47, 5, 33]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(4⁴¹, 44, F₄, 32) (dual of [44, 3, 33]-code), but
        Griesmer bound [i]
      2. linear OA(4⁵, 47, F₄, 3) (dual of [47, 42, 4]-code or 47-cap in PG(4,4)), but
        discarding factors / shortening the dual code would yield linear OA(4⁵, 42, F₄, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,4)), but
        41 is the largest size of a cap in PG(4,Â 4) [i]
  2. OA(4¹², 55, S₄, 8), but
    - discarding factors would yield OA(4¹², 49, S₄, 8), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 17 670136 > 4¹² [i]

(43−32, 43, 59)-Net in Base 4 — Upper bound on s

There is no (11, 43, 60)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁴³, 60, S₄, 32), but
- the linear programming bound shows that MÂ â‰¥ 9 528179 112760 265299 245031 946513 809408 / 118085 180785 > 4⁴³ [i]