MinT - Best Known (256−192, 256, s)-Nets in Base 4

Best Known (256−192, 256, s)-Nets in Base 4

(256−192, 256, 66)-Net over F₄ — Constructive and digital

Digital (64, 256, 66)-net over F₄, using

t-expansion [i] based on digital (49, 256, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(256−192, 256, 99)-Net over F₄ — Digital

Digital (64, 256, 99)-net over F₄, using

t-expansion [i] based on digital (61, 256, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(256−192, 256, 264)-Net over F₄ — Upper bound on s (digital)

There is no digital (64, 256, 265)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁶, 265, F₄, 192) (dual of [265, 9, 193]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4²⁵⁵, 261, F₄, 192) (dual of [261, 6, 193]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(4²⁵⁴, 258, F₄, 192) (dual of [258, 4, 193]-code), but
        Griesmer bound [i]
      2. linear OA(4⁶, 261, F₄, 3) (dual of [261, 255, 4]-code or 261-cap in PG(5,4)), but
        discarding factors / shortening the dual code would yield linear OA(4⁶, 160, F₄, 3) (dual of [160, 154, 4]-code or 160-cap in PG(5,4)), but
        1 times Hill recurrence [i] 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(4⁹, 265, S₄, 4), but
    - discarding factors would yield OA(4⁹, 242, S₄, 4), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 263176 > 4⁹ [i]

(256−192, 256, 415)-Net in Base 4 — Upper bound on s

There is no (64, 256, 416)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14079 882322 618702 627544 673649 273132 143383 124793 409796 514088 489713 687197 268975 215263 651805 029039 706714 529159 752372 145497 721219 359762 551905 156708 549427 308108 > 4²⁵⁶ [i]