MinT - Best Known (45, 56, s)-Nets in Base 81

Best Known (45, 56, s)-Nets in Base 81

(45, 56, 1946802)-Net over F₈₁ — Constructive and digital

Digital (45, 56, 1946802)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (10, 15, 269082)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81¹⁵, 269082, F₈₁, 5, 5) (dual of [(269082, 5), 1345395, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(81¹⁵, 269082, F₈₁, 4, 5) (dual of [(269082, 4), 1076313, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81⁰, s, F₈₁, 4, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81¹, 3322, F₈₁, 4, 1) (dual of [(3322, 4), 13287, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81¹, s, F₈₁, 4, 1) with arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(81¹, 3322, F₈₁, 4, 1) (dual of [(3322, 4), 13287, 2]-NRT-code) (see above)
        linear OOA(81¹, 3322, F₈₁, 4, 1) (dual of [(3322, 4), 13287, 2]-NRT-code) (see above)
        linear OOA(81³, 3322, F₈₁, 4, 2) (dual of [(3322, 4), 13285, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81³, 6643, F₈₁, 4, 2) (dual of [(6643, 4), 26569, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(81³, 6643, F₈₁, 2, 2) (dual of [(6643, 2), 13283, 3]-NRT-code), using
        appending kth column [i] based on linear OA(81³, 6643, F₈₁, 2) (dual of [6643, 6640, 3]-code), using
        Hamming code H(3,81) [i]
        linear OOA(81⁹, 3322, F₈₁, 4, 5) (dual of [(3322, 4), 13279, 6]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(81², 82, F₈₁, 4, 2) (dual of [(82, 4), 326, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;326,81) [i]
        linear OOA(81⁷, 3240, F₈₁, 4, 5) (dual of [(3240, 4), 12953, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (30, 41, 1677720)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴¹, 1677720, F₈₁, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(81⁴¹, 8388601, F₈₁, 11) (dual of [8388601, 8388560, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁴¹, large, F₈₁, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(45, 56, large)-Net over F₈₁ — Digital

Digital (45, 56, large)-net over F₈₁, using

5 times m-reduction [i] based on digital (45, 61, large)-net over F₈₁, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁶¹, large, F₈₁, 16) (dual of [large, large−61, 17]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(45, 56, large)-Net in Base 81 — Upper bound on s

There is no (45, 56, large)-net in base 81, because

9 times m-reduction [i] would yield (45, 47, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (45, 56, s)-Nets in Base 81

(45, 56, 1946802)-Net over F81 — Constructive and digital

(45, 56, large)-Net over F81 — Digital

(45, 56, large)-Net in Base 81 — Upper bound on s

(45, 56, 1946802)-Net over F₈₁ — Constructive and digital

(45, 56, large)-Net over F₈₁ — Digital