MinT - Best Known (21−3, 21, s)-Nets in Base 81

Best Known (21−3, 21, s)-Nets in Base 81

(21−3, 21, large)-Net over F₈₁ — Constructive and digital

Digital (18, 21, large)-net over F₈₁, using

81¹ times duplication [i] based on digital (17, 20, large)-net over F₈₁, using
- t-expansion [i] based on digital (15, 20, large)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81²⁰, large, F₈₁, 5, 5), using
    - appending kth column [i] based on linear OOA(81²⁰, large, F₈₁, 4, 5), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 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, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265721, F₈₁, 4, 0) (dual of [(265721, 4), 1062884, 1]-NRT-code) (see above)
        linear OOA(81¹, 265721, F₈₁, 4, 1) (dual of [(265721, 4), 1062883, 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, 265721, F₈₁, 4, 1) (dual of [(265721, 4), 1062883, 2]-NRT-code) (see above)
        linear OOA(81¹, 265721, F₈₁, 4, 1) (dual of [(265721, 4), 1062883, 2]-NRT-code) (see above)
        linear OOA(81⁴, 265721, F₈₁, 4, 2) (dual of [(265721, 4), 1062880, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81⁴, 538084, F₈₁, 4, 2) (dual of [(538084, 4), 2152332, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(81⁴, 538084, F₈₁, 2, 2) (dual of [(538084, 2), 1076164, 3]-NRT-code), using
        appending kth column [i] based on linear OA(81⁴, 538084, F₈₁, 2) (dual of [538084, 538080, 3]-code), using
        Hamming code H(4,81) [i]
        linear OOA(81¹³, 265721, F₈₁, 4, 5) (dual of [(265721, 4), 1062871, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(81¹³, 531443, F₈₁, 5) (dual of [531443, 531430, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹³, 531444, F₈₁, 5) (dual of [531444, 531431, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(81¹³, 531441, F₈₁, 5) (dual of [531441, 531428, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(81¹⁰, 531441, F₈₁, 4) (dual of [531441, 531431, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(21−3, 21, large)-Net in Base 81 — Upper bound on s

There is no (18, 21, large)-net in base 81, because

1 times m-reduction [i] would yield (18, 20, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (21−3, 21, s)-Nets in Base 81

(21−3, 21, large)-Net over F81 — Constructive and digital

(21−3, 21, large)-Net in Base 81 — Upper bound on s

(21−3, 21, large)-Net over F₈₁ — Constructive and digital