MinT - Best Known (49−21, 49, s)-Nets in Base 27

Best Known (49−21, 49, s)-Nets in Base 27

(49−21, 49, 184)-Net over F₂₇ — Constructive and digital

Digital (28, 49, 184)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 10, 56)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 7, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (4, 14, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
3. digital (4, 25, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(49−21, 49, 250)-Net in Base 27 — Constructive

(28, 49, 250)-net in base 27, using

27¹ times duplication [i] based on (27, 48, 250)-net in base 27, using
- base change [i] based on digital (15, 36, 250)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 11, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
    2. digital (4, 25, 150)-net over F₈₁, using
      - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(49−21, 49, 1038)-Net over F₂₇ — Digital

Digital (28, 49, 1038)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴⁹, 1038, F₂₇, 21) (dual of [1038, 989, 22]-code), using
- 298 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 4 times 0, 1, 14 times 0, 1, 42 times 0, 1, 90 times 0, 1, 142 times 0) [i] based on linear OA(27⁴¹, 732, F₂₇, 21) (dual of [732, 691, 22]-code), using
  - constructionÂ XX applied to C₁Â =Â C([727,18]), C₂Â =Â C([0,19]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,18]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,19]) [i] based on
    1. linear OA(27³⁹, 728, F₂₇, 20) (dual of [728, 689, 21]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,18}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(27³⁹, 728, F₂₇, 20) (dual of [728, 689, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    3. linear OA(27⁴¹, 728, F₂₇, 21) (dual of [728, 687, 22]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,19}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    4. linear OA(27³⁷, 728, F₂₇, 19) (dual of [728, 691, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    5. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
    6. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code) (see above)

(49−21, 49, 1292847)-Net in Base 27 — Upper bound on s

There is no (28, 49, 1292848)-net in base 27, because

1 times m-reduction [i] would yield (28, 48, 1292848)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 507 529785 063463 962656 177370 339908 333708 383418 746678 980866 401484 333601 > 27⁴⁸ [i]

Best Known (49−21, 49, s)-Nets in Base 27

(49−21, 49, 184)-Net over F27 — Constructive and digital

(49−21, 49, 250)-Net in Base 27 — Constructive

(49−21, 49, 1038)-Net over F27 — Digital

(49−21, 49, 1292847)-Net in Base 27 — Upper bound on s

(49−21, 49, 184)-Net over F₂₇ — Constructive and digital

(49−21, 49, 1038)-Net over F₂₇ — Digital