MinT - Best Known (26, 26+3, s)-Nets in Base 128

Best Known (26, 26+3, s)-Nets in Base 128

(26, 26+3, large)-Net over F₁₂₈ — Constructive and digital

Digital (26, 29, large)-net over F₁₂₈, using

3 times m-reduction [i] based on digital (26, 32, large)-net over F₁₂₈, using
- (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 (23, 29, 8388602)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128²⁹, 8388602, F₁₂₈, 9, 6) (dual of [(8388602, 9), 75497389, 7]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(128²⁹, large, F₁₂₈, 3, 6), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128⁰, s, F₁₂₈, 3, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128⁰, 699051, F₁₂₈, 3, 0) (dual of [(699051, 3), 2097153, 1]-NRT-code) (see above)
        linear OOA(128¹, 699051, F₁₂₈, 3, 1) (dual of [(699051, 3), 2097152, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128¹, s, F₁₂₈, 3, 1) with arbitrarily large s, using
        appending 2 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, 699051, F₁₂₈, 3, 1) (dual of [(699051, 3), 2097152, 2]-NRT-code) (see above)
        linear OOA(128¹, 699051, F₁₂₈, 3, 1) (dual of [(699051, 3), 2097152, 2]-NRT-code) (see above)
        linear OOA(128⁴, 699051, F₁₂₈, 3, 2) (dual of [(699051, 3), 2097149, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128⁴, 2113665, F₁₂₈, 3, 2) (dual of [(2113665, 3), 6340991, 3]-NRT-code), using
        appending 1 arbitrary column [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, 699051, F₁₂₈, 3, 3) (dual of [(699051, 3), 2097147, 4]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128⁶, 2130049, F₁₂₈, 3, 3) (dual of [(2130049, 3), 6390141, 4]-NRT-code), using
        OOA stacking with additional row [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(128¹⁶, 699051, F₁₂₈, 3, 6) (dual of [(699051, 3), 2097137, 7]-NRT-code), using
        OOA 3-folding [i] based on linear OA(128¹⁶, 2097153, F₁₂₈, 6) (dual of [2097153, 2097137, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹⁶, 2097155, F₁₂₈, 6) (dual of [2097155, 2097139, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(128¹⁶, 2097152, F₁₂₈, 6) (dual of [2097152, 2097136, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(128¹³, 2097152, F₁₂₈, 5) (dual of [2097152, 2097139, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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]

(26, 26+3, large)-Net in Base 128 — Upper bound on s

There is no (26, 29, large)-net in base 128, because

1 times m-reduction [i] would yield (26, 28, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (26, 26+3, s)-Nets in Base 128

(26, 26+3, large)-Net over F128 — Constructive and digital

(26, 26+3, large)-Net in Base 128 — Upper bound on s

(26, 26+3, large)-Net over F₁₂₈ — Constructive and digital