MinT - Best Known (47, 76, s)-Nets in Base 9

Best Known (47, 76, s)-Nets in Base 9

(47, 76, 344)-Net over F₉ — Constructive and digital

Digital (47, 76, 344)-net over F₉, using

4 times m-reduction [i] based on digital (47, 80, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 40, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(47, 76, 596)-Net over F₉ — Digital

Digital (47, 76, 596)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁷⁶, 596, F₉, 29) (dual of [596, 520, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁷⁶, 728, F₉, 29) (dual of [728, 652, 30]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [0,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(47, 76, 97801)-Net in Base 9 — Upper bound on s

There is no (47, 76, 97802)-net in base 9, because

1 times m-reduction [i] would yield (47, 75, 97802)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 370026 321084 402523 999119 989661 753262 002224 690309 543309 311742 793763 395489 > 9⁷⁵ [i]