MinT - Best Known (102−73, 102, s)-Nets in Base 4

Best Known (102−73, 102, s)-Nets in Base 4

(102−73, 102, 34)-Net over F₄ — Constructive and digital

Digital (29, 102, 34)-net over F₄, using

t-expansion [i] based on digital (21, 102, 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]

(102−73, 102, 42)-Net in Base 4 — Constructive

(29, 102, 42)-net in base 4, using

t-expansion [i] based on (27, 102, 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]

(102−73, 102, 55)-Net over F₄ — Digital

Digital (29, 102, 55)-net over F₄, using

t-expansion [i] based on digital (26, 102, 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]

(102−73, 102, 129)-Net in Base 4 — Upper bound on s

There is no (29, 102, 130)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹⁰², 130, S₄, 73), but
- the linear programming bound shows that MÂ â‰¥ 3 395773 470788 660019 632339 684923 183392 393684 623548 835821 534179 445462 126289 919162 712064 / 129769 154919 419391 372843 > 4¹⁰² [i]