MinT - Best Known (78−13, 78, s)-Nets in Base 81

Best Known (78−13, 78, s)-Nets in Base 81

(78−13, 78, 2802842)-Net over F₈₁ — Constructive and digital

Digital (65, 78, 2802842)-net over F₈₁, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 8, 6642)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁸, 6642, F₈₁, 4, 4) (dual of [(6642, 4), 26560, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(81⁸, 6642, F₈₁, 3, 4) (dual of [(6642, 3), 19918, 5]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81⁰, s, F₈₁, 3, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81⁰, 82, F₈₁, 3, 0) (dual of [(82, 3), 246, 1]-NRT-code) (see above)
        linear OOA(81¹, 82, F₈₁, 3, 1) (dual of [(82, 3), 245, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81¹, s, F₈₁, 3, 1) with arbitrarily large s, using
        appending 2 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, 82, F₈₁, 3, 1) (dual of [(82, 3), 245, 2]-NRT-code) (see above)
        linear OOA(81², 82, F₈₁, 3, 2) (dual of [(82, 3), 244, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;244,81) [i]
        linear OOA(81⁴, 82, F₈₁, 3, 4) (dual of [(82, 3), 242, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;242,81) [i]
2. digital (15, 21, 1398100)-net over F₈₁, using
  - s-reduction based on digital (15, 21, 2796201)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81²¹, 2796201, F₈₁, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(81²¹, large, F₈₁, 6) (dual of [large, large−21, 7]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (36, 49, 1398100)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴⁹, 1398100, F₈₁, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(81⁴⁹, 8388601, F₈₁, 13) (dual of [8388601, 8388552, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁴⁹, large, F₈₁, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(78−13, 78, large)-Net over F₈₁ — Digital

Digital (65, 78, large)-net over F₈₁, using

t-expansion [i] based on digital (60, 78, large)-net over F₈₁, using
- 3 times m-reduction [i] based on digital (60, 81, large)-net over F₈₁, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁸¹, large, F₈₁, 21) (dual of [large, large−81, 22]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(78−13, 78, large)-Net in Base 81 — Upper bound on s

There is no (65, 78, large)-net in base 81, because

11 times m-reduction [i] would yield (65, 67, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (78−13, 78, s)-Nets in Base 81

(78−13, 78, 2802842)-Net over F81 — Constructive and digital

(78−13, 78, large)-Net over F81 — Digital

(78−13, 78, large)-Net in Base 81 — Upper bound on s

(78−13, 78, 2802842)-Net over F₈₁ — Constructive and digital

(78−13, 78, large)-Net over F₈₁ — Digital