MinT - Best Known (78−77, 78, s)-Nets in Base 25

Best Known (78−77, 78, s)-Nets in Base 25

(78−77, 78, 27)-Net over F₂₅ — Constructive and digital

Digital (1, 78, 27)-net over F₂₅, using

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

(78−77, 78, 36)-Net over F₂₅ — Digital

Digital (1, 78, 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]

(78−77, 78, 51)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (1, 78, 52)-net over F₂₅, because

29 times m-reduction [i] would yield digital (1, 49, 52)-net over F₂₅, but
- 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]

(78−77, 78, 52)-Net in Base 25 — Upper bound on s

There is no (1, 78, 53)-net in base 25, because

27 times m-reduction [i] would yield (1, 51, 53)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁵¹, 53, S₂₅, 50), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 4 930380 657631 323783 823303 533017 413935 457540 219431 393779 814243 316650 390625 / 17 > 25⁵¹ [i]