MinT - Best Known (77−61, 77, s)-Nets in Base 4

Best Known (77−61, 77, s)-Nets in Base 4

(77−61, 77, 33)-Net over F₄ — Constructive and digital

Digital (16, 77, 33)-net over F₄, using

t-expansion [i] based on digital (15, 77, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(77−61, 77, 36)-Net over F₄ — Digital

Digital (16, 77, 36)-net over F₄, using

net from sequence [i] based on digital (16, 35)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 16 and N(F) ≥ 36, using
  - Drinfeld modules of rank 1 [i]

(77−61, 77, 72)-Net over F₄ — Upper bound on s (digital)

There is no digital (16, 77, 73)-net over F₄, because

9 times m-reduction [i] would yield digital (16, 68, 73)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁶⁸, 73, F₄, 52) (dual of [73, 5, 53]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4⁶⁷, 70, F₄, 52) (dual of [70, 3, 53]-code), but
      - Griesmer bound [i]
    2. linear OA(4⁵, 73, F₄, 3) (dual of [73, 68, 4]-code or 73-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]

(77−61, 77, 73)-Net in Base 4 — Upper bound on s

There is no (16, 77, 74)-net in base 4, because

13 times m-reduction [i] would yield (16, 64, 74)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁶⁴, 74, S₄, 48), but
  - the linear programming bound shows that MÂ â‰¥ 14291 293180 820859 023858 530456 787498 418848 137216 / 35 673715 > 4⁶⁴ [i]