MinT - Best Known (25, 25+23, s)-Nets in Base 25

Best Known (25, 25+23, s)-Nets in Base 25

(25, 25+23, 200)-Net over F₂₅ — Constructive and digital

Digital (25, 48, 200)-net over F₂₅, using

net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
  - a function field by GarcÃa and Quoos [i]

(25, 25+23, 477)-Net over F₂₅ — Digital

Digital (25, 48, 477)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁴⁸, 477, F₂₅, 23) (dual of [477, 429, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁴⁸, 637, F₂₅, 23) (dual of [637, 589, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,9]) [i] based on
    1. linear OA(25⁴⁵, 626, F₂₅, 23) (dual of [626, 581, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 25⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    2. linear OA(25³⁷, 626, F₂₅, 19) (dual of [626, 589, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 25⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    3. linear OA(25³, 11, F₂₅, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,25) or 11-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]

(25, 25+23, 192223)-Net in Base 25 — Upper bound on s

There is no (25, 48, 192224)-net in base 25, because

1 times m-reduction [i] would yield (25, 47, 192224)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 504878 996385 528066 390619 098314 130989 399170 358847 049524 487795 243777 > 25⁴⁷ [i]