MinT - Best Known (52−27, 52, s)-Nets in Base 25

Best Known (52−27, 52, s)-Nets in Base 25

(52−27, 52, 200)-Net over F₂₅ — Constructive and digital

Digital (25, 52, 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]

(52−27, 52, 315)-Net over F₂₅ — Digital

Digital (25, 52, 315)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25⁵², 315, F₂₅, 2, 27) (dual of [(315, 2), 578, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25⁵², 630, F₂₅, 27) (dual of [630, 578, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(23) [i] based on
    1. linear OA(25⁵⁰, 625, F₂₅, 27) (dual of [625, 575, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    2. linear OA(25⁴⁷, 625, F₂₅, 24) (dual of [625, 578, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    3. linear OA(25², 5, F₂₅, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,25)), using
      - discarding factors / shortening the dual code based on linear OA(25², 25, F₂₅, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
        Reedâ€“Solomon code RS(23,25) [i]

(52−27, 52, 72008)-Net in Base 25 — Upper bound on s

There is no (25, 52, 72009)-net in base 25, because

1 times m-reduction [i] would yield (25, 51, 72009)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 197245 373451 157738 864027 153427 135027 126329 360175 680918 265338 795628 834425 > 25⁵¹ [i]