MinT - Best Known (57, 57+23, s)-Nets in Base 7

Best Known (57, 57+23, s)-Nets in Base 7

(57, 57+23, 226)-Net over F₇ — Constructive and digital

Digital (57, 80, 226)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 12, 26)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 50)-net over F₇, using
      - net defined by OOA [i] based on linear OOA(7⁴, 50, F₇, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(7⁴, 50, F₇, 2, 3) (dual of [(50, 2), 96, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7⁴, 50, F₇, 3) (dual of [50, 46, 4]-code or 50-cap in PG(3,7)), using
        ovoid in PG(3,Â 7) [i]
    2. digital (1, 8, 13)-net over F₇, using
      - 5 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
2. digital (11, 22, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 11, 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 (23, 46, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 23, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(57, 57+23, 2173)-Net over F₇ — Digital

Digital (57, 80, 2173)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸⁰, 2173, F₇, 23) (dual of [2173, 2093, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁸⁰, 2412, F₇, 23) (dual of [2412, 2332, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(19) [i] based on
    1. 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]
    2. linear OA(7⁶⁹, 2401, F₇, 20) (dual of [2401, 2332, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    3. linear OA(7³, 11, F₇, 2) (dual of [11, 8, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(7³, 48, F₇, 2) (dual of [48, 45, 3]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 48Â = 7²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]

(57, 57+23, 959843)-Net in Base 7 — Upper bound on s

There is no (57, 80, 959844)-net in base 7, because

1 times m-reduction [i] would yield (57, 79, 959844)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 5 790942 398881 411806 149643 587586 755800 716961 577613 781600 412250 239905 > 7⁷⁹ [i]