MinT - Best Known (29, 29+69, s)-Nets in Base 4

Best Known (29, 29+69, s)-Nets in Base 4

(29, 29+69, 34)-Net over F₄ — Constructive and digital

Digital (29, 98, 34)-net over F₄, using

t-expansion [i] based on digital (21, 98, 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]

(29, 29+69, 42)-Net in Base 4 — Constructive

(29, 98, 42)-net in base 4, using

t-expansion [i] based on (27, 98, 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]

(29, 29+69, 55)-Net over F₄ — Digital

Digital (29, 98, 55)-net over F₄, using

t-expansion [i] based on digital (26, 98, 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]

(29, 29+69, 140)-Net in Base 4 — Upper bound on s

There is no (29, 98, 141)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁹⁸, 141, S₄, 69), but
- the linear programming bound shows that MÂ â‰¥ 390 667122 461443 438521 906736 556887 277229 974887 246672 391047 184815 799743 048558 730918 115445 571584 / 3696 508035 141223 991028 411517 682665 > 4⁹⁸ [i]