MinT - Best Known (89−26, 89, s)-Nets in Base 7

Best Known (89−26, 89, s)-Nets in Base 7

(89−26, 89, 215)-Net over F₇ — Constructive and digital

Digital (63, 89, 215)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (13, 26, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 13, 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 (28, 54, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 27, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(89−26, 89, 2036)-Net over F₇ — Digital

Digital (63, 89, 2036)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸⁹, 2036, F₇, 26) (dual of [2036, 1947, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁸⁹, 2401, F₇, 26) (dual of [2401, 2312, 27]-code), using
  - an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(89−26, 89, 576660)-Net in Base 7 — Upper bound on s

There is no (63, 89, 576661)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1635 812202 848525 575302 860271 772859 338912 514784 020185 904907 381026 868115 772479 > 7⁸⁹ [i]