MinT - Best Known (17−3, 17, s)-Nets in Base 81

Best Known (17−3, 17, s)-Nets in Base 81

(17−3, 17, large)-Net over F₈₁ — Constructive and digital

Digital (14, 17, large)-net over F₈₁, using

81¹ times duplication [i] based on digital (13, 16, large)-net over F₈₁, using
- t-expansion [i] based on digital (12, 16, large)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81¹⁶, large, F₈₁, 4, 4), using
    - appending kth column [i] based on linear OOA(81¹⁶, large, F₈₁, 3, 4), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 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, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81⁰, 265722, F₈₁, 3, 0) (dual of [(265722, 3), 797166, 1]-NRT-code) (see above)
        linear OOA(81¹, 265722, F₈₁, 3, 1) (dual of [(265722, 3), 797165, 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, 265722, F₈₁, 3, 1) (dual of [(265722, 3), 797165, 2]-NRT-code) (see above)
        linear OOA(81⁴, 265722, F₈₁, 3, 2) (dual of [(265722, 3), 797162, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81⁴, 538084, F₈₁, 3, 2) (dual of [(538084, 3), 1614248, 3]-NRT-code), using
        appending 1 arbitrary column [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(81¹⁰, 265722, F₈₁, 3, 4) (dual of [(265722, 3), 797156, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(81¹⁰, 531444, F₈₁, 4) (dual of [531444, 531434, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(81⁷, 531441, F₈₁, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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]

(17−3, 17, large)-Net in Base 81 — Upper bound on s

There is no (14, 17, large)-net in base 81, because

1 times m-reduction [i] would yield (14, 16, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (17−3, 17, s)-Nets in Base 81

(17−3, 17, large)-Net over F81 — Constructive and digital

(17−3, 17, large)-Net in Base 81 — Upper bound on s

(17−3, 17, large)-Net over F₈₁ — Constructive and digital