MinT - Best Known (64, 86, s)-Nets in Base 7

Best Known (64, 86, s)-Nets in Base 7

(64, 86, 688)-Net over F₇ — Constructive and digital

Digital (64, 86, 688)-net over F₇, using

trace code for nets [i] based on digital (21, 43, 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]

(64, 86, 4806)-Net over F₇ — Digital

Digital (64, 86, 4806)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸⁶, 4806, F₇, 22) (dual of [4806, 4720, 23]-code), using
- trace code [i] based on linear OA(49⁴³, 2403, F₄₉, 22) (dual of [2403, 2360, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
    1. 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]
    2. linear OA(49⁴¹, 2401, F₄₉, 21) (dual of [2401, 2360, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(64, 86, 3311251)-Net in Base 7 — Upper bound on s

There is no (64, 86, 3311252)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 4 769056 466971 706996 292639 729743 155690 167200 318080 808944 162425 612818 806177 > 7⁸⁶ [i]