MinT - Best Known (15, 15+85, s)-Nets in Base 8

Best Known (15, 15+85, s)-Nets in Base 8

(15, 15+85, 65)-Net over F₈ — Constructive and digital

Digital (15, 100, 65)-net over F₈, using

t-expansion [i] based on digital (14, 100, 65)-net over F₈, using
- net from sequence [i] based on digital (14, 64)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
    - the Suzuki function field over F₈ [i]

(15, 15+85, 256)-Net over F₈ — Upper bound on s (digital)

There is no digital (15, 100, 257)-net over F₈, because

extracting embedded orthogonal array [i] would yield linear OA(8¹⁰⁰, 257, F₈, 85) (dual of [257, 157, 86]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(8⁹⁹, 121, S₈, 85), but
    - the linear programming bound shows that MÂ â‰¥ 16 585536 119637 517746 844303 974375 958556 579597 629933 708370 240352 981117 333510 722093 451763 358747 047640 569273 647104 / 63 671965 472816 494885 > 8⁹⁹ [i]
  2. linear OA(8¹⁵⁷, 257, F₈, 136) (dual of [257, 100, 137]-code), but
    - discarding factors / shortening the dual code would yield linear OA(8¹⁵⁷, 238, F₈, 136) (dual of [238, 81, 137]-code), but
      - residual code [i] would yield OA(8²¹, 101, S₈, 17), but
        1 times truncation [i] would yield OA(8²⁰, 100, S₈, 16), but
        the linear programming bound shows that MÂ â‰¥ 766 916854 670851 025894 998613 688320 / 621 810617 698353 > 8²⁰ [i]

(15, 15+85, 288)-Net in Base 8 — Upper bound on s

There is no (15, 100, 289)-net in base 8, because

9 times m-reduction [i] would yield (15, 91, 289)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 15612 322234 848331 880601 230359 684924 233263 993047 381795 871050 142694 117687 977369 078400 > 8⁹¹ [i]