MinT - Best Known (56−21, 56, s)-Nets in Base 27

Best Known (56−21, 56, s)-Nets in Base 27

(56−21, 56, 240)-Net over F₂₇ — Constructive and digital

Digital (35, 56, 240)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 7, 56)-net over F₂₇, using
  - s-reduction based on digital (2, 7, 351)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁷, 351, F₂₇, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]
2. 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)
3. 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]
4. digital (4, 25, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(56−21, 56, 656)-Net in Base 27 — Constructive

(35, 56, 656)-net in base 27, using

base change [i] based on digital (21, 42, 656)-net over F₈₁, using
- 81¹ times duplication [i] based on digital (20, 41, 656)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴¹, 656, F₈₁, 21, 21) (dual of [(656, 21), 13735, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(81⁴¹, 6561, F₈₁, 21) (dual of [6561, 6520, 22]-code), using
      - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(56−21, 56, 3262)-Net over F₂₇ — Digital

Digital (35, 56, 3262)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−21, 56, large)-Net in Base 27 — Upper bound on s

There is no (35, 56, large)-net in base 27, because

19 times m-reduction [i] would yield (35, 37, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (56−21, 56, s)-Nets in Base 27

(56−21, 56, 240)-Net over F27 — Constructive and digital

(56−21, 56, 656)-Net in Base 27 — Constructive

(56−21, 56, 3262)-Net over F27 — Digital

(56−21, 56, large)-Net in Base 27 — Upper bound on s

(56−21, 56, 240)-Net over F₂₇ — Constructive and digital

(56−21, 56, 3262)-Net over F₂₇ — Digital