MinT - Best Known (42−6, 42, s)-Nets in Base 49

Best Known (42−6, 42, s)-Nets in Base 49

(42−6, 42, large)-Net over F₄₉ — Constructive and digital

Digital (36, 42, large)-net over F₄₉, using

t-expansion [i] based on digital (35, 42, large)-net over F₄₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 3, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
      - Niederreiter sequence [i]
  2. digital (32, 39, 8388602)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49³⁹, 8388602, F₄₉, 9, 7) (dual of [(8388602, 9), 75497379, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(49³⁹, large, F₄₉, 3, 7), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(49¹, 1921601, F₄₉, 3, 1) (dual of [(1921601, 3), 5764802, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(49¹, s, F₄₉, 3, 1) with arbitrarily large s, using
        appending 2 arbitrary columns [i] based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(49¹, 1921601, F₄₉, 3, 1) (dual of [(1921601, 3), 5764802, 2]-NRT-code) (see above)
        linear OOA(49⁵, 1921601, F₄₉, 3, 2) (dual of [(1921601, 3), 5764798, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(49⁵, 5884901, F₄₉, 3, 2) (dual of [(5884901, 3), 17654698, 3]-NRT-code), using
        appending 1 arbitrary column [i] based on linear OOA(49⁵, 5884901, F₄₉, 2, 2) (dual of [(5884901, 2), 11769797, 3]-NRT-code), using
        appending kth column [i] based on linear OA(49⁵, 5884901, F₄₉, 2) (dual of [5884901, 5884896, 3]-code), using
        Hamming code H(5,49) [i]
        linear OOA(49⁷, 1921601, F₄₉, 3, 3) (dual of [(1921601, 3), 5764796, 4]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(49⁷, 5769602, F₄₉, 3, 3) (dual of [(5769602, 3), 17308799, 4]-NRT-code), using
        OOA stacking with additional row [i] based on linear OA(49⁷, 5769603, F₄₉, 3) (dual of [5769603, 5769596, 4]-code or 5769603-cap in PG(6,49)), using
        product of two caps with tangent hyperplane [i]
        linear OOA(49²⁵, 1921601, F₄₉, 3, 7) (dual of [(1921601, 3), 5764778, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(49²⁵, 5764803, F₄₉, 7) (dual of [5764803, 5764778, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(49²⁵, 5764805, F₄₉, 7) (dual of [5764805, 5764780, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(49²⁵, 5764801, F₄₉, 7) (dual of [5764801, 5764776, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(49²¹, 5764801, F₄₉, 6) (dual of [5764801, 5764780, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(42−6, 42, large)-Net in Base 49 — Upper bound on s

There is no (36, 42, large)-net in base 49, because

4 times m-reduction [i] would yield (36, 38, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (42−6, 42, s)-Nets in Base 49

(42−6, 42, large)-Net over F49 — Constructive and digital

(42−6, 42, large)-Net in Base 49 — Upper bound on s

(42−6, 42, large)-Net over F₄₉ — Constructive and digital