MinT - Best Known (86−25, 86, s)-Nets in Base 7

Best Known (86−25, 86, s)-Nets in Base 7

(86−25, 86, 216)-Net over F₇ — Constructive and digital

Digital (61, 86, 216)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 12, 16)-net over F₇, using
  - 4 times m-reduction [i] based on digital (4, 16, 16)-net over F₇, using
    - a shift-net [i]
2. digital (12, 24, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 12, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
      - Niederreiter sequence [i]
3. digital (25, 50, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 25, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(86−25, 86, 2073)-Net over F₇ — Digital

Digital (61, 86, 2073)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸⁶, 2073, F₇, 25) (dual of [2073, 1987, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁸⁶, 2410, F₇, 25) (dual of [2410, 2324, 26]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(22) [i] based on
    1. linear OA(7⁸⁵, 2401, F₇, 25) (dual of [2401, 2316, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(7⁷⁷, 2401, F₇, 23) (dual of [2401, 2324, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(7¹, 9, F₇, 1) (dual of [9, 8, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(7¹, s, F₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(86−25, 86, 853723)-Net in Base 7 — Upper bound on s

There is no (61, 86, 853724)-net in base 7, because

1 times m-reduction [i] would yield (61, 85, 853724)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 681292 660191 838341 029053 178058 767929 550447 636952 579710 301275 605418 358529 > 7⁸⁵ [i]