MinT - Best Known (13, 31, s)-Nets in Base 25

Best Known (13, 31, s)-Nets in Base 25

(13, 31, 126)-Net over F₂₅ — Constructive and digital

Digital (13, 31, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 31, 126)-net over F₂₅, using
- net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
    - the Hermitian function field over F₂₅ [i]

(13, 31, 128)-Net over F₂₅ — Digital

Digital (13, 31, 128)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25³¹, 128, F₂₅, 2, 18) (dual of [(128, 2), 225, 19]-NRT-code), using
- constructionÂ X applied to AG(2;F,231P) âŠ‚ AG(2;F,235P) [i] based on
  1. linear OOA(25²⁸, 125, F₂₅, 2, 18) (dual of [(125, 2), 222, 19]-NRT-code), using algebraic-geometric NRT-code AG(2;F,231P) [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
    - the Hermitian function field over F₂₅ [i]
  2. linear OOA(25²⁴, 125, F₂₅, 2, 14) (dual of [(125, 2), 226, 15]-NRT-code), using algebraic-geometric NRT-code AG(2;F,235P) [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126 (see above)
  3. linear OOA(25³, 3, F₂₅, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(25³, 25, F₂₅, 2, 3) (dual of [(25, 2), 47, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;47,25) [i]

(13, 31, 11284)-Net in Base 25 — Upper bound on s

There is no (13, 31, 11285)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 21 685382 446430 003123 182311 471946 211506 909369 > 25³¹ [i]