MinT - Best Known (45−9, 45, s)-Nets in Base 49

Best Known (45−9, 45, s)-Nets in Base 49

(45−9, 45, 2097200)-Net over F₄₉ — Constructive and digital

Digital (36, 45, 2097200)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 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, 41, 2097150)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49⁴¹, 2097150, F₄₉, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(49⁴¹, 8388601, F₄₉, 9) (dual of [8388601, 8388560, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(49⁴¹, large, F₄₉, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 28247525Â | 49¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(45−9, 45, large)-Net over F₄₉ — Digital

Digital (36, 45, large)-net over F₄₉, using

49² times duplication [i] based on digital (34, 43, large)-net over F₄₉, using
- t-expansion [i] based on digital (33, 43, large)-net over F₄₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(45−9, 45, large)-Net in Base 49 — Upper bound on s

There is no (36, 45, large)-net in base 49, because

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