MinT - Best Known (83−22, 83, s)-Nets in Base 27

Best Known (83−22, 83, s)-Nets in Base 27

(83−22, 83, 1875)-Net over F₂₇ — Constructive and digital

Digital (61, 83, 1875)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 19, 86)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 6, 38)-net over F₂₇, using
      - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
    2. digital (2, 13, 48)-net over F₂₇, using
      - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]
2. digital (42, 64, 1789)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶⁴, 1789, F₂₇, 22, 22) (dual of [(1789, 22), 39294, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(27⁶⁴, 19679, F₂₇, 22) (dual of [19679, 19615, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁶⁴, 19683, F₂₇, 22) (dual of [19683, 19619, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(83−22, 83, 1906)-Net in Base 27 — Constructive

(61, 83, 1906)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (7, 18, 116)-net in base 27, using
  - 2 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
    - base change [i] based on digital (2, 15, 116)-net over F₈₁, using
      - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]
2. digital (43, 65, 1790)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶⁵, 1790, F₂₇, 22, 22) (dual of [(1790, 22), 39315, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(27⁶⁵, 19690, F₂₇, 22) (dual of [19690, 19625, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(19) [i] based on
        linear OA(27⁶⁴, 19683, F₂₇, 22) (dual of [19683, 19619, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(27⁵⁸, 19683, F₂₇, 20) (dual of [19683, 19625, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(27¹, 7, F₂₇, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(83−22, 83, 151651)-Net over F₂₇ — Digital

Digital (61, 83, 151651)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−22, 83, large)-Net in Base 27 — Upper bound on s

There is no (61, 83, large)-net in base 27, because

20 times m-reduction [i] would yield (61, 63, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (83−22, 83, s)-Nets in Base 27

(83−22, 83, 1875)-Net over F27 — Constructive and digital

(83−22, 83, 1906)-Net in Base 27 — Constructive

(83−22, 83, 151651)-Net over F27 — Digital

(83−22, 83, large)-Net in Base 27 — Upper bound on s

(83−22, 83, 1875)-Net over F₂₇ — Constructive and digital

(83−22, 83, 151651)-Net over F₂₇ — Digital