MinT - Best Known (88−25, 88, s)-Nets in Base 7

Best Known (88−25, 88, s)-Nets in Base 7

(88−25, 88, 228)-Net over F₇ — Constructive and digital

Digital (63, 88, 228)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 14, 28)-net over F₇, using
  - 1 times m-reduction [i] based on digital (6, 15, 28)-net over F₇, using
    - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) [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)

(88−25, 88, 2424)-Net over F₇ — Digital

Digital (63, 88, 2424)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸⁸, 2424, F₇, 25) (dual of [2424, 2336, 26]-code), using
- 16 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 11 times 0) [i] based on linear OA(7⁸⁵, 2405, F₇, 25) (dual of [2405, 2320, 26]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(23) [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(7⁸¹, 2401, F₇, 24) (dual of [2401, 2320, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    3. linear OA(7⁰, 4, F₇, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(88−25, 88, 1180777)-Net in Base 7 — Upper bound on s

There is no (63, 88, 1180778)-net in base 7, because

1 times m-reduction [i] would yield (63, 87, 1180778)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 33 383370 095408 279393 302506 645398 134682 927107 163117 424182 697658 319734 463257 > 7⁸⁷ [i]