MinT - Best Known (79−22, 79, s)-Nets in Base 7

Best Known (79−22, 79, s)-Nets in Base 7

(79−22, 79, 250)-Net over F₇ — Constructive and digital

Digital (57, 79, 250)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 13, 50)-net over F₇, using
  - base reduction for projective spaces (embedding PG(6,49) in PG(12,7)) for nets [i] based on digital (0, 7, 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 (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 (see above)
3. digital (22, 44, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 22, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(79−22, 79, 2472)-Net over F₇ — Digital

Digital (57, 79, 2472)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁷⁹, 2472, F₇, 22) (dual of [2472, 2393, 23]-code), using
- 65 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 5 times 0, 1, 16 times 0, 1, 39 times 0) [i] based on linear OA(7⁷³, 2401, F₇, 22) (dual of [2401, 2328, 23]-code), using
  - an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(79−22, 79, 959843)-Net in Base 7 — Upper bound on s

There is no (57, 79, 959844)-net in base 7, because

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]