MinT - Best Known (41−3, 41, s)-Nets in Base 49

Best Known (41−3, 41, s)-Nets in Base 49

(41−3, 41, large)-Net over F₄₉ — Constructive and digital

Digital (38, 41, large)-net over F₄₉, using

t-expansion [i] based on digital (37, 41, large)-net over F₄₉, using
- 3 times m-reduction [i] based on digital (37, 44, large)-net over F₄₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (4, 7, 5769603)-net over F₄₉, using
      - net defined by OOA [i] based on linear OOA(49⁷, 5769603, F₄₉, 3, 3) (dual of [(5769603, 3), 17308802, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(49⁷, 5769603, F₄₉, 2, 3) (dual of [(5769603, 2), 11539199, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [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]
    2. digital (30, 37, 5764802)-net over F₄₉, using
      - net defined by OOA [i] based on linear OOA(49³⁷, 5764802, F₄₉, 9, 7) (dual of [(5764802, 9), 51883181, 8]-NRT-code), using
        OOA stacking with additional row [i] based on linear OOA(49³⁷, 5764803, F₄₉, 3, 7) (dual of [(5764803, 3), 17294372, 8]-NRT-code), using
        generalized (u,Â u+v)-construction [i] based on
        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)) (see above)
        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]

(41−3, 41, large)-Net in Base 49 — Upper bound on s

There is no (38, 41, large)-net in base 49, because

1 times m-reduction [i] would yield (38, 40, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (41−3, 41, s)-Nets in Base 49

(41−3, 41, large)-Net over F49 — Constructive and digital

(41−3, 41, large)-Net in Base 49 — Upper bound on s

(41−3, 41, large)-Net over F₄₉ — Constructive and digital