MinT - Best Known (70−19, 70, s)-Nets in Base 7

Best Known (70−19, 70, s)-Nets in Base 7

(70−19, 70, 302)-Net over F₇ — Constructive and digital

Digital (51, 70, 302)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 12, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 6, 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]
2. digital (9, 18, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 9, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
3. digital (21, 40, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 20, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(70−19, 70, 2581)-Net over F₇ — Digital

Digital (51, 70, 2581)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁷⁰, 2581, F₇, 19) (dual of [2581, 2511, 20]-code), using
- 171 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 4 times 0, 1, 16 times 0, 1, 45 times 0, 1, 102 times 0) [i] based on linear OA(7⁶⁵, 2405, F₇, 19) (dual of [2405, 2340, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
    1. linear OA(7⁶⁵, 2401, F₇, 19) (dual of [2401, 2336, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. linear OA(7⁶¹, 2401, F₇, 18) (dual of [2401, 2340, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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]

(70−19, 70, 2082974)-Net in Base 7 — Upper bound on s

There is no (51, 70, 2082975)-net in base 7, because

1 times m-reduction [i] would yield (51, 69, 2082975)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 20500 528588 197032 269274 542252 209125 245532 323207 503985 189771 > 7⁶⁹ [i]