MinT - Best Known (128−99, 128, s)-Nets in Base 4

Best Known (128−99, 128, s)-Nets in Base 4

(128−99, 128, 34)-Net over F₄ — Constructive and digital

Digital (29, 128, 34)-net over F₄, using

t-expansion [i] based on digital (21, 128, 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]

(128−99, 128, 42)-Net in Base 4 — Constructive

(29, 128, 42)-net in base 4, using

t-expansion [i] based on (27, 128, 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]

(128−99, 128, 55)-Net over F₄ — Digital

Digital (29, 128, 55)-net over F₄, using

t-expansion [i] based on digital (26, 128, 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]

(128−99, 128, 122)-Net in Base 4 — Upper bound on s

There is no (29, 128, 123)-net in base 4, because

19 times m-reduction [i] would yield (29, 109, 123)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁰⁹, 123, S₄, 80), but
  - the linear programming bound shows that MÂ â‰¥ 64 104606 306644 111085 844977 797492 841266 115540 167873 142109 538693 549498 826752 / 127 155457 > 4¹⁰⁹ [i]