MinT - Best Known (257−194, 257, s)-Nets in Base 4

Best Known (257−194, 257, s)-Nets in Base 4

(257−194, 257, 66)-Net over F₄ — Constructive and digital

Digital (63, 257, 66)-net over F₄, using

t-expansion [i] based on digital (49, 257, 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]

(257−194, 257, 99)-Net over F₄ — Digital

Digital (63, 257, 99)-net over F₄, using

t-expansion [i] based on digital (61, 257, 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]

(257−194, 257, 260)-Net over F₄ — Upper bound on s (digital)

There is no digital (63, 257, 261)-net over F₄, because

2 times m-reduction [i] would yield digital (63, 255, 261)-net over F₄, but
- extracting embedded orthogonal array [i] would yield 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]

(257−194, 257, 408)-Net in Base 4 — Upper bound on s

There is no (63, 257, 409)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 60915 451826 100071 622953 550118 801466 482243 200085 972584 897928 008016 319604 458060 677992 115816 471642 866839 337068 423347 223766 652328 537985 611151 638387 401325 380240 > 4²⁵⁷ [i]