MinT - Best Known (57, 81, s)-Nets in Base 7

Best Known (57, 81, s)-Nets in Base 7

(57, 81, 213)-Net over F₇ — Constructive and digital

Digital (57, 81, 213)-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 (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 (24, 48, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 24, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(57, 81, 1772)-Net over F₇ — Digital

Digital (57, 81, 1772)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸¹, 1772, F₇, 24) (dual of [1772, 1691, 25]-code), using
- discarding factors / shortening the dual code based on 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]

(57, 81, 446287)-Net in Base 7 — Upper bound on s

There is no (57, 81, 446288)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 283 758013 068950 445034 766261 973998 254287 936076 152920 498700 521041 262209 > 7⁸¹ [i]