MinT - Best Known (48−23, 48, s)-Nets in Base 27

Best Known (48−23, 48, s)-Nets in Base 27

(48−23, 48, 164)-Net over F₂₇ — Constructive and digital

Digital (25, 48, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 18, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (7, 30, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(48−23, 48, 224)-Net in Base 27 — Constructive

(25, 48, 224)-net in base 27, using

base change [i] based on digital (13, 36, 224)-net over F₈₁, using
- net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
    - a function field by SÃ©mirat [i]

(48−23, 48, 524)-Net over F₂₇ — Digital

Digital (25, 48, 524)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴⁸, 524, F₂₇, 23) (dual of [524, 476, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁴⁸, 741, F₂₇, 23) (dual of [741, 693, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,9]) [i] based on
    1. linear OA(27⁴⁵, 730, F₂₇, 23) (dual of [730, 685, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    2. linear OA(27³⁷, 730, F₂₇, 19) (dual of [730, 693, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    3. linear OA(27³, 11, F₂₇, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-cap in PG(2,27)), using
      - discarding factors / shortening the dual code based on linear OA(27³, 27, F₂₇, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
        Reedâ€“Solomon code RS(24,27) [i]

(48−23, 48, 246523)-Net in Base 27 — Upper bound on s

There is no (25, 48, 246524)-net in base 27, because

1 times m-reduction [i] would yield (25, 47, 246524)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 18 797818 945308 119037 184847 731262 955519 478165 659831 101194 542311 191665 > 27⁴⁷ [i]