MinT - Best Known (99−70, 99, s)-Nets in Base 4

Best Known (99−70, 99, s)-Nets in Base 4

(99−70, 99, 34)-Net over F₄ — Constructive and digital

Digital (29, 99, 34)-net over F₄, using

t-expansion [i] based on digital (21, 99, 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]

(99−70, 99, 42)-Net in Base 4 — Constructive

(29, 99, 42)-net in base 4, using

t-expansion [i] based on (27, 99, 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]

(99−70, 99, 55)-Net over F₄ — Digital

Digital (29, 99, 55)-net over F₄, using

t-expansion [i] based on digital (26, 99, 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]

(99−70, 99, 137)-Net in Base 4 — Upper bound on s

There is no (29, 99, 138)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁹⁹, 138, S₄, 70), but
- the linear programming bound shows that MÂ â‰¥ 118246 438673 063111 509987 411413 863923 581254 775008 190429 184491 048612 519959 485941 226484 592604 610560 / 288411 558856 539326 726871 768616 117917 > 4⁹⁹ [i]