MinT - Best Known (49−48, 49, s)-Nets in Base 25

Best Known (49−48, 49, s)-Nets in Base 25

(49−48, 49, 27)-Net over F₂₅ — Constructive and digital

Digital (1, 49, 27)-net over F₂₅, using

net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
- Niederreiter sequence [i]

(49−48, 49, 36)-Net over F₂₅ — Digital

Digital (1, 49, 36)-net over F₂₅, using

net from sequence [i] based on digital (1, 35)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 1 and N(F) ≥ 36, using
  - sharp bound on the number of rational points on curves with genus 1 [i]

(49−48, 49, 51)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (1, 49, 52)-net over F₂₅, because

extracting embedded orthogonal array [i] would yield linear OA(25⁴⁹, 52, F₂₅, 48) (dual of [52, 3, 49]-code), but
- bound for almost-MDS-codes with dimension nÂ =Â 3 [i]

(49−48, 49, 93)-Net in Base 25 — Upper bound on s

There is no (1, 49, 94)-net in base 25, because

11 times m-reduction [i] would yield (1, 38, 94)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25³⁸, 94, S₂₅, 37), but
  - the linear programming bound shows that MÂ â‰¥ 789627 539185 406277 823242 593427 721658 372320 234775 543212 890625 / 5 958181 > 25³⁸ [i]