MinT - Best Known (64, 64+194, s)-Nets in Base 4

Best Known (64, 64+194, s)-Nets in Base 4

(64, 64+194, 66)-Net over F₄ — Constructive and digital

Digital (64, 258, 66)-net over F₄, using

t-expansion [i] based on digital (49, 258, 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]

(64, 64+194, 99)-Net over F₄ — Digital

Digital (64, 258, 99)-net over F₄, using

t-expansion [i] based on digital (61, 258, 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]

(64, 64+194, 264)-Net over F₄ — Upper bound on s (digital)

There is no digital (64, 258, 265)-net over F₄, because

2 times m-reduction [i] would yield digital (64, 256, 265)-net over F₄, but
- 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
        linear OA(4²⁵⁴, 258, F₄, 192) (dual of [258, 4, 193]-code), but
        Griesmer bound [i]
        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]

(64, 64+194, 415)-Net in Base 4 — Upper bound on s

There is no (64, 258, 416)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 247827 731690 416674 431767 132849 469876 731415 643779 215178 153415 444681 555365 121940 917040 270318 847271 086224 108072 439811 169602 336741 931143 324603 351991 936589 310412 > 4²⁵⁸ [i]