MinT - Best Known (42, 42+25, s)-Nets in Base 9

Best Known (42, 42+25, s)-Nets in Base 9

(42, 42+25, 344)-Net over F₉ — Constructive and digital

Digital (42, 67, 344)-net over F₉, using

3 times m-reduction [i] based on digital (42, 70, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 35, 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]

(42, 42+25, 633)-Net over F₉ — Digital

Digital (42, 67, 633)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁶⁷, 633, F₉, 25) (dual of [633, 566, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁶⁷, 728, F₉, 25) (dual of [728, 661, 26]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [0,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(42, 42+25, 117106)-Net in Base 9 — Upper bound on s

There is no (42, 67, 117107)-net in base 9, because

1 times m-reduction [i] would yield (42, 66, 117107)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 955 099565 406407 174103 716747 447104 817418 372486 813887 357573 263905 > 9⁶⁶ [i]