MinT - Best Known (85−65, 85, s)-Nets in Base 4

Best Known (85−65, 85, s)-Nets in Base 4

(85−65, 85, 33)-Net over F₄ — Constructive and digital

Digital (20, 85, 33)-net over F₄, using

t-expansion [i] based on digital (15, 85, 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]

(85−65, 85, 41)-Net over F₄ — Digital

Digital (20, 85, 41)-net over F₄, using

t-expansion [i] based on digital (18, 85, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(85−65, 85, 87)-Net over F₄ — Upper bound on s (digital)

There is no digital (20, 85, 88)-net over F₄, because

1 times m-reduction [i] would yield digital (20, 84, 88)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁸⁴, 88, F₄, 64) (dual of [88, 4, 65]-code), but
  - Griesmer bound [i]

(85−65, 85, 88)-Net in Base 4 — Upper bound on s

There is no (20, 85, 89)-net in base 4, because

6 times m-reduction [i] would yield (20, 79, 89)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁷⁹, 89, S₄, 59), but
  - the linear programming bound shows that MÂ â‰¥ 61920 901370 435694 838453 718992 523478 976782 549980 807168 / 142375 > 4⁷⁹ [i]