MinT - Best Known (15, 30, s)-Nets in Base 25

Best Known (15, 30, s)-Nets in Base 25

(15, 30, 132)-Net over F₂₅ — Constructive and digital

Digital (15, 30, 132)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 11, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]
2. digital (4, 19, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅ (see above)

(15, 30, 315)-Net over F₂₅ — Digital

Digital (15, 30, 315)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25³⁰, 315, F₂₅, 2, 15) (dual of [(315, 2), 600, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25³⁰, 630, F₂₅, 15) (dual of [630, 600, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(25³⁰, 631, F₂₅, 15) (dual of [631, 601, 16]-code), using
    - constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
      1. linear OA(25²⁹, 626, F₂₅, 15) (dual of [626, 597, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 25⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
      2. linear OA(25²⁵, 626, F₂₅, 13) (dual of [626, 601, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 25⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (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]

(15, 30, 87127)-Net in Base 25 — Upper bound on s

There is no (15, 30, 87128)-net in base 25, because

1 times m-reduction [i] would yield (15, 29, 87128)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 34695 026212 631899 136104 035097 750650 949825 > 25²⁹ [i]