MinT - Best Known (85−68, 85, s)-Nets in Base 4

Best Known (85−68, 85, s)-Nets in Base 4

(85−68, 85, 33)-Net over F₄ — Constructive and digital

Digital (17, 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]

(85−68, 85, 40)-Net over F₄ — Digital

Digital (17, 85, 40)-net over F₄, using

net from sequence [i] based on digital (17, 39)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 17 and N(F) ≥ 40, using
  - Drinfeld modules of rank 1 [i]

(85−68, 85, 76)-Net over F₄ — Upper bound on s (digital)

There is no digital (17, 85, 77)-net over F₄, because

15 times m-reduction [i] would yield digital (17, 70, 77)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁷⁰, 77, F₄, 53) (dual of [77, 7, 54]-code), but
  - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(85−68, 85, 77)-Net in Base 4 — Upper bound on s

There is no (17, 85, 78)-net in base 4, because

17 times m-reduction [i] would yield (17, 68, 78)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁶⁸, 78, S₄, 51), but
  - the linear programming bound shows that MÂ â‰¥ 28277 344911 736830 143467 290769 718122 389583 167488 / 244673 > 4⁶⁸ [i]