MinT - Best Known (51−49, 51, s)-Nets in Base 25

Best Known (51−49, 51, s)-Nets in Base 25

(51−49, 51, 28)-Net over F₂₅ — Constructive and digital

Digital (2, 51, 28)-net over F₂₅, using

net from sequence [i] based on digital (2, 27)-sequence over F₂₅, using
- Niederreiter sequence [i]

(51−49, 51, 46)-Net over F₂₅ — Digital

Digital (2, 51, 46)-net over F₂₅, using

net from sequence [i] based on digital (2, 45)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 2 and N(F) ≥ 46, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(51−49, 51, 101)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (2, 51, 102)-net over F₂₅, because

1 times m-reduction [i] would yield digital (2, 50, 102)-net over F₂₅, but
- extracting embedded orthogonal array [i] would yield linear OA(25⁵⁰, 102, F₂₅, 48) (dual of [102, 52, 49]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(25⁴⁹, 52, F₂₅, 48) (dual of [52, 3, 49]-code), but
      - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
    2. linear OA(25⁵², 102, F₂₅, 50) (dual of [102, 50, 51]-code), but
      - discarding factors / shortening the dual code would yield linear OA(25⁵², 78, F₂₅, 50) (dual of [78, 26, 51]-code), but
        residual code [i] would yield OA(25², 27, S₂₅, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 649 > 25² [i]

(51−49, 51, 186)-Net in Base 25 — Upper bound on s

There is no (2, 51, 187)-net in base 25, because

6 times m-reduction [i] would yield (2, 45, 187)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁴⁵, 187, S₂₅, 43), but
  - the linear programming bound shows that MÂ â‰¥ 111 488601 007381 568900 361473 791104 508011 644280 696117 448314 907960 593700 408935 546875 / 137824 248946 330259 > 25⁴⁵ [i]