MinT - Best Known (50−11, 50, s)-Nets in Base 49

Best Known (50−11, 50, s)-Nets in Base 49

(50−11, 50, 1154186)-Net over F₄₉ — Constructive and digital

Digital (39, 50, 1154186)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 9, 1226)-net over F₄₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 50)-net over F₄₉, using
      - kÂ =Â 2 construction [i]
    2. digital (2, 7, 1176)-net over F₄₉, using
      - net defined by OOA [i] based on linear OOA(49⁷, 1176, F₄₉, 5, 5) (dual of [(1176, 5), 5873, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (30, 41, 1152960)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49⁴¹, 1152960, F₄₉, 11, 11) (dual of [(1152960, 11), 12682519, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(49⁴¹, 5764801, F₄₉, 11) (dual of [5764801, 5764760, 12]-code), using
      - an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(50−11, 50, large)-Net over F₄₉ — Digital

Digital (39, 50, large)-net over F₄₉, using

49² times duplication [i] based on digital (37, 48, large)-net over F₄₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(50−11, 50, large)-Net in Base 49 — Upper bound on s

There is no (39, 50, large)-net in base 49, because

9 times m-reduction [i] would yield (39, 41, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]