MinT - Best Known (98−27, 98, s)-Nets in Base 9

Best Known (98−27, 98, s)-Nets in Base 9

(98−27, 98, 740)-Net over F₉ — Constructive and digital

Digital (71, 98, 740)-net over F₉, using

12 times m-reduction [i] based on digital (71, 110, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 55, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(98−27, 98, 6398)-Net over F₉ — Digital

Digital (71, 98, 6398)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁹⁸, 6398, F₉, 27) (dual of [6398, 6300, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁹⁸, 6571, F₉, 27) (dual of [6571, 6473, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(9⁹⁷, 6562, F₉, 27) (dual of [6562, 6465, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 9⁸âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    2. linear OA(9⁸⁹, 6562, F₉, 25) (dual of [6562, 6473, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 9⁸âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(9¹, 9, F₉, 1) (dual of [9, 8, 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]

(98−27, 98, large)-Net in Base 9 — Upper bound on s

There is no (71, 98, large)-net in base 9, because

25 times m-reduction [i] would yield (71, 73, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]