MinT - Best Known (98−83, 98, s)-Nets in Base 8

Best Known (98−83, 98, s)-Nets in Base 8

(98−83, 98, 65)-Net over F₈ — Constructive and digital

Digital (15, 98, 65)-net over F₈, using

t-expansion [i] based on digital (14, 98, 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]

(98−83, 98, 271)-Net over F₈ — Upper bound on s (digital)

There is no digital (15, 98, 272)-net over F₈, because

3 times m-reduction [i] would yield digital (15, 95, 272)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8⁹⁵, 272, F₈, 80) (dual of [272, 177, 81]-code), but
  - residual code [i] would yield OA(8¹⁵, 191, S₈, 10), but
    - the linear programming bound shows that MÂ â‰¥ 25 427098 382149 761175 126016 / 717528 710051 > 8¹⁵ [i]

(98−83, 98, 288)-Net in Base 8 — Upper bound on s

There is no (15, 98, 289)-net in base 8, because

7 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]