MinT - Best Known (31−3, 31, s)-Nets in Base 9

Best Known (31−3, 31, s)-Nets in Base 9

(31−3, 31, large)-Net over F₉ — Constructive and digital

Digital (28, 31, large)-net over F₉, using

9³ times duplication [i] based on digital (25, 28, large)-net over F₉, using
- t-expansion [i] based on digital (24, 28, large)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9²⁸, large, F₉, 4, 4), using
    - appending kth column [i] based on linear OOA(9²⁸, large, F₉, 3, 4), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(9², 10, F₉, 3, 2) (dual of [(10, 3), 28, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;28,9) [i]
        linear OOA(9²⁶, 8388602, F₉, 3, 4) (dual of [(8388602, 3), 25165780, 5]-NRT-code), using
        trace code [i] based on linear OOA(81¹³, 4194301, F₈₁, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(81¹³, 8388602, F₈₁, 4) (dual of [8388602, 8388589, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹³, large, F₈₁, 4) (dual of [large, large−13, 5]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(31−3, 31, large)-Net in Base 9 — Upper bound on s

There is no (28, 31, large)-net in base 9, because

1 times m-reduction [i] would yield (28, 30, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]