MinT - Best Known (108−79, 108, s)-Nets in Base 4

Best Known (108−79, 108, s)-Nets in Base 4

(108−79, 108, 34)-Net over F₄ — Constructive and digital

Digital (29, 108, 34)-net over F₄, using

t-expansion [i] based on digital (21, 108, 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]

(108−79, 108, 42)-Net in Base 4 — Constructive

(29, 108, 42)-net in base 4, using

t-expansion [i] based on (27, 108, 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]

(108−79, 108, 55)-Net over F₄ — Digital

Digital (29, 108, 55)-net over F₄, using

t-expansion [i] based on digital (26, 108, 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]

(108−79, 108, 123)-Net in Base 4 — Upper bound on s

There is no (29, 108, 124)-net in base 4, because

1 times m-reduction [i] would yield (29, 107, 124)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁰⁷, 124, S₄, 78), but
  - the linear programming bound shows that MÂ â‰¥ 49019 846676 977689 436041 401684 986717 032746 670105 891391 278865 372850 212766 220288 / 1 444154 771455 > 4¹⁰⁷ [i]