MinT - Best Known (33−17, 33, s)-Nets in Base 27

Best Known (33−17, 33, s)-Nets in Base 27

(33−17, 33, 128)-Net over F₂₇ — Constructive and digital

Digital (16, 33, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 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]
2. digital (4, 21, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(33−17, 33, 172)-Net in Base 27 — Constructive

(16, 33, 172)-net in base 27, using

3 times m-reduction [i] based on (16, 36, 172)-net in base 27, using
- base change [i] based on digital (7, 27, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(33−17, 33, 337)-Net over F₂₇ — Digital

Digital (16, 33, 337)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(27³³, 337, F₂₇, 2, 17) (dual of [(337, 2), 641, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(27³³, 366, F₂₇, 2, 17) (dual of [(366, 2), 699, 18]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(27³³, 732, F₂₇, 17) (dual of [732, 699, 18]-code), using
    - constructionÂ XX applied to C₁Â =Â C([727,14]), C₂Â =Â C([0,15]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,14]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,15]) [i] based on
      1. linear OA(27³¹, 728, F₂₇, 16) (dual of [728, 697, 17]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,14}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      2. linear OA(27³¹, 728, F₂₇, 16) (dual of [728, 697, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      3. linear OA(27³³, 728, F₂₇, 17) (dual of [728, 695, 18]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,15}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
      4. linear OA(27²⁹, 728, F₂₇, 15) (dual of [728, 699, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [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)

(33−17, 33, 76939)-Net in Base 27 — Upper bound on s

There is no (16, 33, 76940)-net in base 27, because

1 times m-reduction [i] would yield (16, 32, 76940)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6362 908124 916560 761557 535080 172656 048018 504129 > 27³² [i]

Best Known (33−17, 33, s)-Nets in Base 27

(33−17, 33, 128)-Net over F27 — Constructive and digital

(33−17, 33, 172)-Net in Base 27 — Constructive

(33−17, 33, 337)-Net over F27 — Digital

(33−17, 33, 76939)-Net in Base 27 — Upper bound on s

(33−17, 33, 128)-Net over F₂₇ — Constructive and digital

(33−17, 33, 337)-Net over F₂₇ — Digital