MinT - Best Known (127−3, 127, s)-Nets in Base 9

Best Known (127−3, 127, s)-Nets in Base 9

(127−3, 127, large)-Net over F₉ — Constructive and digital

Digital (124, 127, large)-net over F₉, using

t-expansion [i] based on digital (122, 127, large)-net over F₉, using
- 6 times m-reduction [i] based on digital (122, 133, large)-net over F₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 10)-net over F₉, using
      - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
        the rational function field F₉(x) [i]
        Niederreiter sequence [i]
    2. digital (117, 128, 8388602)-net over F₉, using
      - trace code for nets [i] based on digital (53, 64, 4194301)-net over F₈₁, using
        net defined by OOA [i] based on linear OOA(81⁶⁴, 4194301, F₈₁, 15, 11) (dual of [(4194301, 15), 62914451, 12]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(81⁶⁴, large, F₈₁, 3, 11), 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(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(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(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(81⁴, 177148, F₈₁, 3, 2) (dual of [(177148, 3), 531440, 3]-NRT-code) (see above)
        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(81¹³, 177148, F₈₁, 3, 5) (dual of [(177148, 3), 531431, 6]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81¹³, 265721, F₈₁, 3, 5) (dual of [(265721, 3), 797150, 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]
        linear OOA(81³¹, 177148, F₈₁, 3, 11) (dual of [(177148, 3), 531413, 12]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81³¹, 531444, F₈₁, 11) (dual of [531444, 531413, 12]-code), using
        constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
        linear OA(81³¹, 531441, F₈₁, 11) (dual of [531441, 531410, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(81²⁸, 531441, F₈₁, 10) (dual of [531441, 531413, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)

(127−3, 127, large)-Net in Base 9 — Upper bound on s

There is no (124, 127, large)-net in base 9, because

1 times m-reduction [i] would yield (124, 126, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (127−3, 127, s)-Nets in Base 9

(127−3, 127, large)-Net over F9 — Constructive and digital

(127−3, 127, large)-Net in Base 9 — Upper bound on s

(127−3, 127, large)-Net over F₉ — Constructive and digital