MinT - Best Known (53, 53+9, s)-Nets in Base 128

Best Known (53, 53+9, s)-Nets in Base 128

(53, 53+9, large)-Net over F₁₂₈ — Constructive and digital

Digital (53, 62, large)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 13, 4194301)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128¹³, 4194301, F₁₂₈, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(128¹³, 8388602, F₁₂₈, 4) (dual of [8388602, 8388589, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(128¹³, large, F₁₂₈, 4) (dual of [large, large−13, 5]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 17895697Â | 128⁴âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
2. digital (40, 49, 7242231)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴⁹, 7242231, F₁₂₈, 12, 9) (dual of [(7242231, 12), 86906723, 10]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(128⁴⁹, 7242232, F₁₂₈, 4, 9) (dual of [(7242232, 4), 28968879, 10]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 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, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128⁰, 49156, F₁₂₈, 4, 0) (dual of [(49156, 4), 196624, 1]-NRT-code) (see above)
        linear OOA(128¹, 49156, F₁₂₈, 4, 1) (dual of [(49156, 4), 196623, 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, 49156, F₁₂₈, 4, 1) (dual of [(49156, 4), 196623, 2]-NRT-code) (see above)
        linear OOA(128¹, 49156, F₁₂₈, 4, 1) (dual of [(49156, 4), 196623, 2]-NRT-code) (see above)
        linear OOA(128¹, 49156, F₁₂₈, 4, 1) (dual of [(49156, 4), 196623, 2]-NRT-code) (see above)
        linear OOA(128¹, 49156, F₁₂₈, 4, 1) (dual of [(49156, 4), 196623, 2]-NRT-code) (see above)
        linear OOA(128⁴, 49156, F₁₂₈, 4, 2) (dual of [(49156, 4), 196620, 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(128⁵, 49156, F₁₂₈, 4, 3) (dual of [(49156, 4), 196619, 4]-NRT-code), using
        appending 1 arbitrary column [i] based on linear OOA(128⁵, 49156, F₁₂₈, 3, 3) (dual of [(49156, 3), 147463, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(128⁵, 49156, F₁₂₈, 2, 3) (dual of [(49156, 2), 98307, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁵, 49156, F₁₂₈, 3) (dual of [49156, 49151, 4]-code or 49156-cap in PG(4,128)), using
        large caps in PG(4,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) with 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]

(53, 53+9, large)-Net in Base 128 — Upper bound on s

There is no (53, 62, large)-net in base 128, because

7 times m-reduction [i] would yield (53, 55, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (53, 53+9, s)-Nets in Base 128

(53, 53+9, large)-Net over F128 — Constructive and digital

(53, 53+9, large)-Net in Base 128 — Upper bound on s

(53, 53+9, large)-Net over F₁₂₈ — Constructive and digital