MinT - Best Known (57−20, 57, s)-Nets in Base 128

Best Known (57−20, 57, s)-Nets in Base 128

(57−20, 57, 2025)-Net over F₁₂₈ — Constructive and digital

Digital (37, 57, 2025)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 387)-net over F₁₂₈, using
  - 1 times m-reduction [i] based on digital (8, 19, 387)-net over F₁₂₈, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 3, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
      2. digital (0, 5, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
      3. digital (0, 11, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
2. digital (19, 39, 1638)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128³⁹, 1638, F₁₂₈, 20, 20) (dual of [(1638, 20), 32721, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(128³⁹, 16380, F₁₂₈, 20) (dual of [16380, 16341, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(128³⁹, 16384, F₁₂₈, 20) (dual of [16384, 16345, 21]-code), using
        an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(57−20, 57, 6810)-Net in Base 128 — Constructive

(37, 57, 6810)-net in base 128, using

128¹ times duplication [i] based on (36, 56, 6810)-net in base 128, using
- base change [i] based on digital (29, 49, 6810)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 10, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
    2. digital (19, 39, 6553)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256³⁹, 6553, F₂₅₆, 20, 20) (dual of [(6553, 20), 131021, 21]-NRT-code), using
        OA 10-folding and stacking [i] based on linear OA(256³⁹, 65530, F₂₅₆, 20) (dual of [65530, 65491, 21]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁹, 65536, F₂₅₆, 20) (dual of [65536, 65497, 21]-code), using
        an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(57−20, 57, 130940)-Net over F₁₂₈ — Digital

Digital (37, 57, 130940)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−20, 57, large)-Net in Base 128 — Upper bound on s

There is no (37, 57, large)-net in base 128, because

18 times m-reduction [i] would yield (37, 39, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (57−20, 57, s)-Nets in Base 128

(57−20, 57, 2025)-Net over F128 — Constructive and digital

(57−20, 57, 6810)-Net in Base 128 — Constructive

(57−20, 57, 130940)-Net over F128 — Digital

(57−20, 57, large)-Net in Base 128 — Upper bound on s

(57−20, 57, 2025)-Net over F₁₂₈ — Constructive and digital

(57−20, 57, 130940)-Net over F₁₂₈ — Digital