MinT - Best Known (3, 3+69, s)-Nets in Base 25

Best Known (3, 3+69, s)-Nets in Base 25

(3, 3+69, 52)-Net over F₂₅ — Constructive and digital

Digital (3, 72, 52)-net over F₂₅, using

net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
  - a function field by GarcÃa and Quoos [i]

(3, 3+69, 56)-Net over F₂₅ — Digital

Digital (3, 72, 56)-net over F₂₅, using

net from sequence [i] based on digital (3, 55)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 56, using
  - fields by Serre with genus 3 or 4 [i]

(3, 3+69, 147)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (3, 72, 148)-net over F₂₅, because

2 times m-reduction [i] would yield digital (3, 70, 148)-net over F₂₅, but
- extracting embedded orthogonal array [i] would yield linear OA(25⁷⁰, 148, F₂₅, 67) (dual of [148, 78, 68]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(25⁶⁹, 73, S₂₅, 67), but
      - the linear programming bound shows that MÂ â‰¥ 24178 564222 846123 668546 399721 917528 895562 151649 914153 732205 158997 548011 257094 913162 291049 957275 390625 / 7242 > 25⁶⁹ [i]
    2. linear OA(25⁷⁸, 148, F₂₅, 75) (dual of [148, 70, 76]-code), but
      - discarding factors / shortening the dual code would yield linear OA(25⁷⁸, 103, F₂₅, 75) (dual of [103, 25, 76]-code), but
        residual code [i] would yield OA(25³, 27, S₂₅, 3), but
        bound for OAs with index unity [i]

(3, 3+69, 216)-Net in Base 25 — Upper bound on s

There is no (3, 72, 217)-net in base 25, because

extracting embedded orthogonal array [i] would yield OA(25⁷², 217, S₂₅, 69), but
- the linear programming bound shows that MÂ â‰¥ 355519 391662 314672 380570 021829 487766 144872 565794 937458 458152 614928 942211 677704 104640 963811 423404 195011 244155 466556 549072 265625 / 7 857365 392716 571699 898392 > 25⁷² [i]