MinT - Best Known (57−27, 57, s)-Nets in Base 25

Best Known (57−27, 57, s)-Nets in Base 25

(57−27, 57, 204)-Net over F₂₅ — Constructive and digital

Digital (30, 57, 204)-net over F₂₅, using

net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
  - a function field by GarcÃa and Quoos [i]

(57−27, 57, 562)-Net over F₂₅ — Digital

Digital (30, 57, 562)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁵⁷, 562, F₂₅, 27) (dual of [562, 505, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁵⁷, 645, F₂₅, 27) (dual of [645, 588, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(18) [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(25³⁷, 625, F₂₅, 19) (dual of [625, 588, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(25⁷, 20, F₂₅, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,25)), using
      - discarding factors / shortening the dual code based on linear OA(25⁷, 25, F₂₅, 7) (dual of [25, 18, 8]-code or 25-arc in PG(6,25)), using
        Reedâ€“Solomon code RS(18,25) [i]

(57−27, 57, 248358)-Net in Base 25 — Upper bound on s

There is no (30, 57, 248359)-net in base 25, because

1 times m-reduction [i] would yield (30, 56, 248359)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 1 925986 613982 674108 345735 723461 742055 964164 030466 134126 185175 363276 767971 920265 > 25⁵⁶ [i]