MinT - Best Known (13, 30, s)-Nets in Base 25

Best Known (13, 30, s)-Nets in Base 25

(13, 30, 126)-Net over F₂₅ — Constructive and digital

Digital (13, 30, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 30, 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, 30, 129)-Net over F₂₅ — Digital

Digital (13, 30, 129)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25³⁰, 129, F₂₅, 17) (dual of [129, 99, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25³⁰, 132, F₂₅, 17) (dual of [132, 102, 18]-code), using
  - constructionÂ X applied to AG(F,107P) âŠ‚ AG(F,111P) [i] based on
    1. linear OA(25²⁷, 125, F₂₅, 17) (dual of [125, 98, 18]-code), using algebraic-geometric code AG(F,107P) [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 OA(25²³, 125, F₂₅, 13) (dual of [125, 102, 14]-code), using algebraic-geometric code AG(F,111P) [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126 (see above)
    3. linear OA(25³, 7, F₂₅, 3) (dual of [7, 4, 4]-code or 7-arc in PG(2,25) or 7-cap in PG(2,25)), using
      - discarding factors / shortening the dual code based on linear OA(25³, 25, F₂₅, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
        Reedâ€“Solomon code RS(22,25) [i]

(13, 30, 18319)-Net in Base 25 — Upper bound on s

There is no (13, 30, 18320)-net in base 25, because

1 times m-reduction [i] would yield (13, 29, 18320)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 34697 647774 611258 094434 688563 873495 376897 > 25²⁹ [i]