MinT - Best Known (39, 39+23, s)-Nets in Base 9

Best Known (39, 39+23, s)-Nets in Base 9

(39, 39+23, 344)-Net over F₉ — Constructive and digital

Digital (39, 62, 344)-net over F₉, using

2 times m-reduction [i] based on digital (39, 64, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 32, 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]

(39, 39+23, 630)-Net over F₉ — Digital

Digital (39, 62, 630)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁶², 630, F₉, 23) (dual of [630, 568, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁶², 737, F₉, 23) (dual of [737, 675, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(9⁶¹, 730, F₉, 23) (dual of [730, 669, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 9⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    2. linear OA(9⁵⁵, 730, F₉, 21) (dual of [730, 675, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 9⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    3. linear OA(9¹, 7, F₉, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(39, 39+23, 120118)-Net in Base 9 — Upper bound on s

There is no (39, 62, 120119)-net in base 9, because

1 times m-reduction [i] would yield (39, 61, 120119)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 16173 963377 864732 345194 916231 070964 779792 784446 221279 779305 > 9⁶¹ [i]