MinT - Best Known (27−4, 27, s)-Nets in Base 9

Best Known (27−4, 27, s)-Nets in Base 9

(27−4, 27, 8388602)-Net over F₉ — Constructive and digital

Digital (23, 27, 8388602)-net over F₉, using

9¹ times duplication [i] based on digital (22, 26, 8388602)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9²⁶, 8388602, F₉, 4, 4) (dual of [(8388602, 4), 33554382, 5]-NRT-code), using
  - appending kth column [i] based on 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]

(27−4, 27, large)-Net over F₉ — Digital

Digital (23, 27, large)-net over F₉, using

9³ times duplication [i] based on digital (20, 24, large)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9²⁴, large, F₉, 4, 4), using
  - appending kth column [i] based on linear OOA(9²⁴, large, F₉, 3, 4), using
    - Gilbertâ€“VarÅ¡amov bound [i]

(27−4, 27, large)-Net in Base 9 — Upper bound on s

There is no (23, 27, large)-net in base 9, because

2 times m-reduction [i] would yield (23, 25, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]