MinT - Best Known (54−27, 54, s)-Nets in Base 25

Best Known (54−27, 54, s)-Nets in Base 25

(54−27, 54, 200)-Net over F₂₅ — Constructive and digital

Digital (27, 54, 200)-net over F₂₅, using

t-expansion [i] based on digital (25, 54, 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]

(54−27, 54, 379)-Net over F₂₅ — Digital

Digital (27, 54, 379)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁵⁴, 379, F₂₅, 27) (dual of [379, 325, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁵⁴, 631, F₂₅, 27) (dual of [631, 577, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(25⁵³, 626, F₂₅, 27) (dual of [626, 573, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 25⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    2. linear OA(25⁴⁹, 626, F₂₅, 25) (dual of [626, 577, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 25⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(25¹, 5, F₂₅, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(54−27, 54, 118158)-Net in Base 25 — Upper bound on s

There is no (27, 54, 118159)-net in base 25, because

1 times m-reduction [i] would yield (27, 53, 118159)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 123 267013 186866 921931 742069 610644 703508 998016 784407 937263 982016 012103 322185 > 25⁵³ [i]