MinT - Best Known (31−5, 31, s)-Nets in Base 81

Best Known (31−5, 31, s)-Nets in Base 81

(31−5, 31, large)-Net over F₈₁ — Constructive and digital

Digital (26, 31, large)-net over F₈₁, using

1 times m-reduction [i] based on digital (26, 32, large)-net over F₈₁, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 3, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
      - Niederreiter sequence [i]
  2. digital (23, 29, 8388602)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81²⁹, 8388602, F₈₁, 9, 6) (dual of [(8388602, 9), 75497389, 7]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(81²⁹, large, F₈₁, 3, 6), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 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, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81⁰, 177148, F₈₁, 3, 0) (dual of [(177148, 3), 531444, 1]-NRT-code) (see above)
        linear OOA(81¹, 177148, F₈₁, 3, 1) (dual of [(177148, 3), 531443, 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, 177148, F₈₁, 3, 1) (dual of [(177148, 3), 531443, 2]-NRT-code) (see above)
        linear OOA(81¹, 177148, F₈₁, 3, 1) (dual of [(177148, 3), 531443, 2]-NRT-code) (see above)
        linear OOA(81⁴, 177148, F₈₁, 3, 2) (dual of [(177148, 3), 531440, 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(81⁶, 177148, F₈₁, 3, 3) (dual of [(177148, 3), 531438, 4]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81⁶, 538247, F₈₁, 3, 3) (dual of [(538247, 3), 1614735, 4]-NRT-code), using
        OOA stacking with additional row [i] based on linear OA(81⁶, 538248, F₈₁, 3) (dual of [538248, 538242, 4]-code or 538248-cap in PG(5,81)), using
        large caps in PG(5,Â b) [i]
        linear OOA(81¹⁶, 177148, F₈₁, 3, 6) (dual of [(177148, 3), 531428, 7]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81¹⁶, 531444, F₈₁, 6) (dual of [531444, 531428, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(81¹⁶, 531441, F₈₁, 6) (dual of [531441, 531425, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        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(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]

(31−5, 31, large)-Net in Base 81 — Upper bound on s

There is no (26, 31, large)-net in base 81, because

3 times m-reduction [i] would yield (26, 28, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (31−5, 31, s)-Nets in Base 81

(31−5, 31, large)-Net over F81 — Constructive and digital

(31−5, 31, large)-Net in Base 81 — Upper bound on s

(31−5, 31, large)-Net over F₈₁ — Constructive and digital