MinT - Best Known (76−20, 76, s)-Nets in Base 7

Best Known (76−20, 76, s)-Nets in Base 7

(76−20, 76, 304)-Net over F₇ — Constructive and digital

Digital (56, 76, 304)-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 (12, 22, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 11, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]
3. digital (22, 42, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 21, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(76−20, 76, 3187)-Net over F₇ — Digital

Digital (56, 76, 3187)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁷⁶, 3187, F₇, 20) (dual of [3187, 3111, 21]-code), using
- 775 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 25 times 0, 1, 71 times 0, 1, 147 times 0, 1, 229 times 0, 1, 292 times 0) [i] based on linear OA(7⁶⁹, 2405, F₇, 20) (dual of [2405, 2336, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
    1. 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]
    2. 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]
    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]

(76−20, 76, 1997877)-Net in Base 7 — Upper bound on s

There is no (56, 76, 1997878)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 16883 112791 313201 433728 138475 200386 831140 186685 297991 102402 148909 > 7⁷⁶ [i]