MinT - Best Known (82−64, 82, s)-Nets in Base 4

Best Known (82−64, 82, s)-Nets in Base 4

(82−64, 82, 33)-Net over F₄ — Constructive and digital

Digital (18, 82, 33)-net over F₄, using

t-expansion [i] based on digital (15, 82, 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]

(82−64, 82, 41)-Net over F₄ — Digital

Digital (18, 82, 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]

(82−64, 82, 80)-Net over F₄ — Upper bound on s (digital)

There is no digital (18, 82, 81)-net over F₄, because

8 times m-reduction [i] would yield digital (18, 74, 81)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁷⁴, 81, F₄, 56) (dual of [81, 7, 57]-code), but
  - residual code [i] would yield linear OA(4¹⁸, 24, F₄, 14) (dual of [24, 6, 15]-code), but
    - â€œBouâ€ bound on codes from Brouwerâ€™s database [i]

(82−64, 82, 81)-Net in Base 4 — Upper bound on s

There is no (18, 82, 82)-net in base 4, because

10 times m-reduction [i] would yield (18, 72, 82)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁷², 82, S₄, 54), but
  - the linear programming bound shows that MÂ â‰¥ 258 868477 512236 179386 827675 994871 629856 557051 674624 / 8 133191 > 4⁷² [i]