MinT - Best Known (42−10, 42, s)-Nets in Base 49

Best Known (42−10, 42, s)-Nets in Base 49

(42−10, 42, 1153011)-Net over F₄₉ — Constructive and digital

Digital (32, 42, 1153011)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 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 (27, 37, 1152961)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49³⁷, 1152961, F₄₉, 10, 10) (dual of [(1152961, 10), 11529573, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(49³⁷, 5764805, F₄₉, 10) (dual of [5764805, 5764768, 11]-code), using
      - constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
        linear OA(49³⁷, 5764801, F₄₉, 10) (dual of [5764801, 5764764, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(49³³, 5764801, F₄₉, 9) (dual of [5764801, 5764768, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [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]

(42−10, 42, 6669484)-Net over F₄₉ — Digital

Digital (32, 42, 6669484)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−10, 42, large)-Net in Base 49 — Upper bound on s

There is no (32, 42, large)-net in base 49, because

8 times m-reduction [i] would yield (32, 34, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (42−10, 42, s)-Nets in Base 49

(42−10, 42, 1153011)-Net over F49 — Constructive and digital

(42−10, 42, 6669484)-Net over F49 — Digital

(42−10, 42, large)-Net in Base 49 — Upper bound on s

(42−10, 42, 1153011)-Net over F₄₉ — Constructive and digital

(42−10, 42, 6669484)-Net over F₄₉ — Digital