MinT - Best Known (33−10, 33, s)-Nets in Base 49

Best Known (33−10, 33, s)-Nets in Base 49

(33−10, 33, 23580)-Net over F₄₉ — Constructive and digital

Digital (23, 33, 23580)-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 (18, 28, 23530)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49²⁸, 23530, F₄₉, 10, 10) (dual of [(23530, 10), 235272, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(49²⁸, 117650, F₄₉, 10) (dual of [117650, 117622, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(49²⁸, 117652, F₄₉, 10) (dual of [117652, 117624, 11]-code), using
        constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
        linear OA(49²⁸, 117649, F₄₉, 10) (dual of [117649, 117621, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 117648Â = 49³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(49²⁵, 117649, F₄₉, 9) (dual of [117649, 117624, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 117648Â = 49³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(49⁰, 3, F₄₉, 0) (dual of [3, 3, 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]

(33−10, 33, 136116)-Net over F₄₉ — Digital

Digital (23, 33, 136116)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(33−10, 33, large)-Net in Base 49 — Upper bound on s

There is no (23, 33, large)-net in base 49, because

8 times m-reduction [i] would yield (23, 25, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (33−10, 33, s)-Nets in Base 49

(33−10, 33, 23580)-Net over F49 — Constructive and digital

(33−10, 33, 136116)-Net over F49 — Digital

(33−10, 33, large)-Net in Base 49 — Upper bound on s

(33−10, 33, 23580)-Net over F₄₉ — Constructive and digital

(33−10, 33, 136116)-Net over F₄₉ — Digital