MinT - Best Known (38, 38+112, s)-Nets in Base 4

Best Known (38, 38+112, s)-Nets in Base 4

(38, 38+112, 56)-Net over F₄ — Constructive and digital

Digital (38, 150, 56)-net over F₄, using

t-expansion [i] based on digital (33, 150, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(38, 38+112, 66)-Net over F₄ — Digital

Digital (38, 150, 66)-net over F₄, using

t-expansion [i] based on digital (37, 150, 66)-net over F₄, using
- net from sequence [i] based on digital (37, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
    - Drinfeld modules of rank 1 [i]

(38, 38+112, 164)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 150, 165)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁰, 165, F₄, 112) (dual of [165, 15, 113]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4¹⁴⁹, 157, F₄, 112) (dual of [157, 8, 113]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(4¹⁴⁸, 153, F₄, 112) (dual of [153, 5, 113]-code), but
        residual code [i] would yield linear OA(4³⁶, 40, F₄, 28) (dual of [40, 4, 29]-code), but
        residual code [i] would yield linear OA(4⁸, 11, F₄, 7) (dual of [11, 3, 8]-code), but
        improvement by Maruta on the Griesmer bound [i]
      2. OA(4⁸, 157, S₄, 4), but
        discarding factors would yield OA(4⁸, 121, S₄, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 65704 > 4⁸ [i]
  2. OA(4¹⁵, 165, S₄, 8), but
    - discarding factors would yield OA(4¹⁵, 135, S₄, 8), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1082 768311 > 4¹⁵ [i]

(38, 38+112, 252)-Net in Base 4 — Upper bound on s

There is no (38, 150, 253)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 043360 558714 911151 593248 059152 083109 597226 052479 991358 887140 178795 107904 400403 532522 361845 > 4¹⁵⁰ [i]