MinT - Best Known (33−7, 33, s)-Nets in Base 9

Best Known (33−7, 33, s)-Nets in Base 9

(33−7, 33, 19687)-Net over F₉ — Constructive and digital

Digital (26, 33, 19687)-net over F₉, using

9¹ times duplication [i] based on digital (25, 32, 19687)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9³², 19687, F₉, 7, 7) (dual of [(19687, 7), 137777, 8]-NRT-code), using
  - OOA 3-folding and stacking with additional row [i] based on linear OA(9³², 59062, F₉, 7) (dual of [59062, 59030, 8]-code), using
    - constructionÂ X4 applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
      1. linear OA(9³¹, 59050, F₉, 7) (dual of [59050, 59019, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 9¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
      2. linear OA(9²¹, 59050, F₉, 5) (dual of [59050, 59029, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 9¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
      3. linear OA(9¹¹, 12, F₉, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,9)), using
        dual of repetition code with length 12 [i]
      4. linear OA(9¹, 12, F₉, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(9¹, 728, F₉, 1) (dual of [728, 727, 2]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [0,0], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]

(33−7, 33, 66296)-Net over F₉ — Digital

Digital (26, 33, 66296)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(33−7, 33, large)-Net in Base 9 — Upper bound on s

There is no (26, 33, large)-net in base 9, because

5 times m-reduction [i] would yield (26, 28, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]