MinT - Best Known (45, 45+25, s)-Nets in Base 128

Best Known (45, 45+25, s)-Nets in Base 128

(45, 45+25, 1686)-Net over F₁₂₈ — Constructive and digital

Digital (45, 70, 1686)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 21, 321)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 6, 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 (3, 15, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]
2. digital (24, 49, 1365)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴⁹, 1365, F₁₂₈, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(128⁴⁹, 16381, F₁₂₈, 25) (dual of [16381, 16332, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴⁹, 16384, F₁₂₈, 25) (dual of [16384, 16335, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(45, 45+25, 5718)-Net in Base 128 — Constructive

(45, 70, 5718)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 14, 257)-net in base 128, using
  - 2 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
    - base change [i] based on digital (0, 14, 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. (31, 56, 5461)-net in base 128, using
  - base change [i] based on digital (24, 49, 5461)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁹, 5461, F₂₅₆, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
      - OOA 12-folding and stacking with additional row [i] based on linear OA(256⁴⁹, 65533, F₂₅₆, 25) (dual of [65533, 65484, 26]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, 65536, F₂₅₆, 25) (dual of [65536, 65487, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(45, 45+25, 108053)-Net over F₁₂₈ — Digital

Digital (45, 70, 108053)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+25, large)-Net in Base 128 — Upper bound on s

There is no (45, 70, large)-net in base 128, because

23 times m-reduction [i] would yield (45, 47, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (45, 45+25, s)-Nets in Base 128

(45, 45+25, 1686)-Net over F128 — Constructive and digital

(45, 45+25, 5718)-Net in Base 128 — Constructive

(45, 45+25, 108053)-Net over F128 — Digital

(45, 45+25, large)-Net in Base 128 — Upper bound on s

(45, 45+25, 1686)-Net over F₁₂₈ — Constructive and digital

(45, 45+25, 108053)-Net over F₁₂₈ — Digital