MinT - Best Known (45−5, 45, s)-Nets in Base 128

Best Known (45−5, 45, s)-Nets in Base 128

(45−5, 45, large)-Net over F₁₂₈ — Constructive and digital

Digital (40, 45, large)-net over F₁₂₈, using

3 times m-reduction [i] based on digital (40, 48, large)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 4, 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 (36, 44, 8388602)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁴⁴, 8388602, F₁₂₈, 12, 8) (dual of [(8388602, 12), 100663180, 9]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(128⁴⁴, large, F₁₂₈, 4, 8), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128⁰, s, F₁₂₈, 4, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128⁰, 524288, F₁₂₈, 4, 0) (dual of [(524288, 4), 2097152, 1]-NRT-code) (see above)
        linear OOA(128¹, 524288, F₁₂₈, 4, 1) (dual of [(524288, 4), 2097151, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128¹, s, F₁₂₈, 4, 1) with arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(128¹, 524288, F₁₂₈, 4, 1) (dual of [(524288, 4), 2097151, 2]-NRT-code) (see above)
        linear OOA(128¹, 524288, F₁₂₈, 4, 1) (dual of [(524288, 4), 2097151, 2]-NRT-code) (see above)
        linear OOA(128¹, 524288, F₁₂₈, 4, 1) (dual of [(524288, 4), 2097151, 2]-NRT-code) (see above)
        linear OOA(128⁴, 524288, F₁₂₈, 4, 2) (dual of [(524288, 4), 2097148, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128⁴, 2113665, F₁₂₈, 4, 2) (dual of [(2113665, 4), 8454656, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(128⁴, 2113665, F₁₂₈, 2, 2) (dual of [(2113665, 2), 4227326, 3]-NRT-code), using
        appending kth column [i] based on linear OA(128⁴, 2113665, F₁₂₈, 2) (dual of [2113665, 2113661, 3]-code), using
        Hamming code H(4,128) [i]
        linear OOA(128⁴, 524288, F₁₂₈, 4, 2) (dual of [(524288, 4), 2097148, 3]-NRT-code) (see above)
        linear OOA(128¹⁰, 524288, F₁₂₈, 4, 4) (dual of [(524288, 4), 2097142, 5]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128¹⁰, 1048577, F₁₂₈, 4, 4) (dual of [(1048577, 4), 4194298, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(128¹⁰, 2097154, F₁₂₈, 4) (dual of [2097154, 2097144, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹⁰, 2097155, F₁₂₈, 4) (dual of [2097155, 2097145, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(128¹⁰, 2097152, F₁₂₈, 4) (dual of [2097152, 2097142, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(128⁷, 2097152, F₁₂₈, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
        linear OOA(128²², 524288, F₁₂₈, 4, 8) (dual of [(524288, 4), 2097130, 9]-NRT-code), using
        OOA 4-folding [i] based on linear OA(128²², 2097152, F₁₂₈, 8) (dual of [2097152, 2097130, 9]-code), using
        an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(45−5, 45, large)-Net in Base 128 — Upper bound on s

There is no (40, 45, large)-net in base 128, because

3 times m-reduction [i] would yield (40, 42, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (45−5, 45, s)-Nets in Base 128

(45−5, 45, large)-Net over F128 — Constructive and digital

(45−5, 45, large)-Net in Base 128 — Upper bound on s

(45−5, 45, large)-Net over F₁₂₈ — Constructive and digital