MinT - Best Known (27, 27+111, s)-Nets in Base 4

Best Known (27, 27+111, s)-Nets in Base 4

(27, 27+111, 34)-Net over F₄ — Constructive and digital

Digital (27, 138, 34)-net over F₄, using

t-expansion [i] based on digital (21, 138, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(27, 27+111, 42)-Net in Base 4 — Constructive

(27, 138, 42)-net in base 4, using

net from sequence [i] based on (27, 41)-sequence in base 4, using
- base expansion [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(27, 27+111, 55)-Net over F₄ — Digital

Digital (27, 138, 55)-net over F₄, using

t-expansion [i] based on digital (26, 138, 55)-net over F₄, using
- net from sequence [i] based on digital (26, 54)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
    - Drinfeld modules of rank 1 [i]

(27, 27+111, 115)-Net in Base 4 — Upper bound on s

There is no (27, 138, 116)-net in base 4, because

36 times m-reduction [i] would yield (27, 102, 116)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁰², 116, S₄, 75), but
  - the linear programming bound shows that MÂ â‰¥ 173566 451646 449122 083079 076441 789443 549958 833470 789796 847564 973808 812032 / 4774 621397 > 4¹⁰² [i]