MinT - Best Known (83−20, 83, s)-Nets in Base 9

Best Known (83−20, 83, s)-Nets in Base 9

(83−20, 83, 1313)-Net over F₉ — Constructive and digital

Digital (63, 83, 1313)-net over F₉, using

1 times m-reduction [i] based on digital (63, 84, 1313)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁸⁴, 1313, F₉, 21, 21) (dual of [(1313, 21), 27489, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(9⁸⁴, 13131, F₉, 21) (dual of [13131, 13047, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁸⁴, 13134, F₉, 21) (dual of [13134, 13050, 22]-code), using
      - trace code [i] based on linear OA(81⁴², 6567, F₈₁, 21) (dual of [6567, 6525, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(81⁴¹, 6562, F₈₁, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(81³⁷, 6562, F₈₁, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(81¹, 5, F₈₁, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(83−20, 83, 14620)-Net over F₉ — Digital

Digital (63, 83, 14620)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−20, 83, large)-Net in Base 9 — Upper bound on s

There is no (63, 83, large)-net in base 9, because

18 times m-reduction [i] would yield (63, 65, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]