MinT - Best Known (34, 34+19, s)-Nets in Base 9

Best Known (34, 34+19, s)-Nets in Base 9

(34, 34+19, 344)-Net over F₉ — Constructive and digital

Digital (34, 53, 344)-net over F₉, using

1 times m-reduction [i] based on digital (34, 54, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 27, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(34, 34+19, 735)-Net over F₉ — Digital

Digital (34, 53, 735)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁵³, 735, F₉, 19) (dual of [735, 682, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁵³, 742, F₉, 19) (dual of [742, 689, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(14) [i] based on
    1. linear OA(9⁴⁹, 729, F₉, 19) (dual of [729, 680, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. linear OA(9⁴⁰, 729, F₉, 15) (dual of [729, 689, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    3. linear OA(9⁴, 13, F₉, 3) (dual of [13, 9, 4]-code or 13-cap in PG(3,9)), using
      - discarding factors / shortening the dual code based on linear OA(9⁴, 82, F₉, 3) (dual of [82, 78, 4]-code or 82-cap in PG(3,9)), using
        ovoid in PG(3,Â 9) [i]

(34, 34+19, 169063)-Net in Base 9 — Upper bound on s

There is no (34, 53, 169064)-net in base 9, because

1 times m-reduction [i] would yield (34, 52, 169064)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 41 747030 966683 317299 030935 005464 789249 947439 673665 > 9⁵² [i]