MinT - Information on Result #3178594

Information on Result #3178594

Digital (45, 51, large)-net over F₁₂₈, using 3 times m-reduction based on digital (45, 54, 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 (41, 50, 8388602)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁵⁰, 8388602, F₁₂₈, 12, 9) (dual of [(8388602, 12), 100663174, 10]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(128⁵⁰, large, F₁₂₈, 4, 9), 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) for 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(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) for arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) for 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(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(128⁶, 524288, F₁₂₈, 4, 3) (dual of [(524288, 4), 2097146, 4]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128⁶, 2130050, F₁₂₈, 4, 3) (dual of [(2130050, 4), 8520194, 4]-NRT-code), using
        appending 1 arbitrary column [i] based on linear OOA(128⁶, 2130050, F₁₂₈, 3, 3) (dual of [(2130050, 3), 6390144, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(128⁶, 2130050, F₁₂₈, 2, 3) (dual of [(2130050, 2), 4260094, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁶, 2130050, F₁₂₈, 3) (dual of [2130050, 2130044, 4]-code or 2130050-cap in PG(5,128)), using
        large caps in PG(5,Â b) [i]
        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) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
        linear OOA(128²⁵, 524288, F₁₂₈, 4, 9) (dual of [(524288, 4), 2097127, 10]-NRT-code), using
        OOA 4-folding [i] based on linear OA(128²⁵, 2097152, F₁₂₈, 9) (dual of [2097152, 2097127, 10]-code), using
        an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, k and t, m and t.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		This result only	Method
1	Digital (46, 51, large)-net over F₁₂₈	[i]		t-Expansion
2	Digital (47, 51, large)-net over F₁₂₈	[i]		t-Expansion