MinT - Best Known (67, 67+23, s)-Nets in Base 7

Best Known (67, 67+23, s)-Nets in Base 7

(67, 67+23, 688)-Net over F₇ — Constructive and digital

Digital (67, 90, 688)-net over F₇, using

2 times m-reduction [i] based on digital (67, 92, 688)-net over F₇, using
- trace code for nets [i] based on digital (21, 46, 344)-net over F₄₉, using
  - net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
      - the Hermitian function field over F₄₉ [i]

(67, 67+23, 4806)-Net over F₇ — Digital

Digital (67, 90, 4806)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁹⁰, 4806, F₇, 23) (dual of [4806, 4716, 24]-code), using
- trace code [i] based on linear OA(49⁴⁵, 2403, F₄₉, 23) (dual of [2403, 2358, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
    1. linear OA(49⁴⁵, 2401, F₄₉, 23) (dual of [2401, 2356, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(49⁴³, 2401, F₄₉, 22) (dual of [2401, 2358, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(49⁰, 2, F₄₉, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(67, 67+23, 5629551)-Net in Base 7 — Upper bound on s

There is no (67, 90, 5629552)-net in base 7, because

1 times m-reduction [i] would yield (67, 89, 5629552)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1635 784243 263281 337506 026291 878532 902737 324808 401390 827010 636959 403432 192257 > 7⁸⁹ [i]