MinT - Best Known (33, 33+13, s)-Nets in Base 27

Best Known (33, 33+13, s)-Nets in Base 27

(33, 33+13, 3336)-Net over F₂₇ — Constructive and digital

Digital (33, 46, 3336)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 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, 6, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (24, 37, 3280)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27³⁷, 3280, F₂₇, 13, 13) (dual of [(3280, 13), 42603, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(27³⁷, 19681, F₂₇, 13) (dual of [19681, 19644, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(27³⁷, 19683, F₂₇, 13) (dual of [19683, 19646, 14]-code), using
        an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(33, 33+13, 3363)-Net in Base 27 — Constructive

(33, 46, 3363)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (2, 8, 82)-net in base 27, using
  - base change [i] based on digital (0, 6, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
      - Niederreiter sequence [i]
2. digital (25, 38, 3281)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27³⁸, 3281, F₂₇, 13, 13) (dual of [(3281, 13), 42615, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(27³⁸, 19687, F₂₇, 13) (dual of [19687, 19649, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(27³⁸, 19691, F₂₇, 13) (dual of [19691, 19653, 14]-code), using
        constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
        linear OA(27³⁷, 19684, F₂₇, 13) (dual of [19684, 19647, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(27³¹, 19684, F₂₇, 11) (dual of [19684, 19653, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [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]

(33, 33+13, 62420)-Net over F₂₇ — Digital

Digital (33, 46, 62420)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(33, 33+13, large)-Net in Base 27 — Upper bound on s

There is no (33, 46, large)-net in base 27, because

11 times m-reduction [i] would yield (33, 35, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (33, 33+13, s)-Nets in Base 27

(33, 33+13, 3336)-Net over F27 — Constructive and digital

(33, 33+13, 3363)-Net in Base 27 — Constructive

(33, 33+13, 62420)-Net over F27 — Digital

(33, 33+13, large)-Net in Base 27 — Upper bound on s

(33, 33+13, 3336)-Net over F₂₇ — Constructive and digital

(33, 33+13, 62420)-Net over F₂₇ — Digital