MinT - Best Known (76−20, 76, s)-Nets in Base 27

Best Known (76−20, 76, s)-Nets in Base 27

(76−20, 76, 2054)-Net over F₂₇ — Constructive and digital

Digital (56, 76, 2054)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 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, 12, 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 (38, 58, 1968)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁵⁸, 1968, F₂₇, 20, 20) (dual of [(1968, 20), 39302, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(27⁵⁸, 19680, F₂₇, 20) (dual of [19680, 19622, 21]-code), using
      - discarding factors / shortening the dual code based on 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]

(76−20, 76, 2085)-Net in Base 27 — Constructive

(56, 76, 2085)-net in base 27, using

27¹ times duplication [i] based on (55, 75, 2085)-net in base 27, using
- (u,Â u+v)-construction [i] based on
  1. (6, 16, 116)-net in base 27, using
    - base change [i] based on digital (2, 12, 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 (39, 59, 1969)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁵⁹, 1969, F₂₇, 20, 20) (dual of [(1969, 20), 39321, 21]-NRT-code), using
      - OA 10-folding and stacking [i] based on linear OA(27⁵⁹, 19690, F₂₇, 20) (dual of [19690, 19631, 21]-code), using
        constructionÂ X applied to C_e(19) âŠ‚ C_e(17) [i] based on
        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(27⁵², 19683, F₂₇, 18) (dual of [19683, 19631, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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]

(76−20, 76, 162077)-Net over F₂₇ — Digital

Digital (56, 76, 162077)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(76−20, 76, large)-Net in Base 27 — Upper bound on s

There is no (56, 76, large)-net in base 27, because

18 times m-reduction [i] would yield (56, 58, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (76−20, 76, s)-Nets in Base 27

(76−20, 76, 2054)-Net over F27 — Constructive and digital

(76−20, 76, 2085)-Net in Base 27 — Constructive

(76−20, 76, 162077)-Net over F27 — Digital

(76−20, 76, large)-Net in Base 27 — Upper bound on s

(76−20, 76, 2054)-Net over F₂₇ — Constructive and digital

(76−20, 76, 162077)-Net over F₂₇ — Digital