MinT - Best Known (68−19, 68, s)-Nets in Base 7

Best Known (68−19, 68, s)-Nets in Base 7

(68−19, 68, 300)-Net over F₇ — Constructive and digital

Digital (49, 68, 300)-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 (19, 38, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 19, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(68−19, 68, 2430)-Net over F₇ — Digital

Digital (49, 68, 2430)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁶⁸, 2430, F₇, 19) (dual of [2430, 2362, 20]-code), using
- 22 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 4 times 0, 1, 16 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]

(68−19, 68, 1351710)-Net in Base 7 — Upper bound on s

There is no (49, 68, 1351711)-net in base 7, because

1 times m-reduction [i] would yield (49, 67, 1351711)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 418 377920 243679 540420 517455 439358 467593 097275 395949 342859 > 7⁶⁷ [i]