MinT - Best Known (29−8, 29, s)-Nets in Base 7

Best Known (29−8, 29, s)-Nets in Base 7

(29−8, 29, 608)-Net over F₇ — Constructive and digital

Digital (21, 29, 608)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 8)-net over F₇, using
  - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
      - the rational function field F₇(x) [i]
    - Niederreiter sequence [i]
2. digital (17, 25, 600)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7²⁵, 600, F₇, 8, 8) (dual of [(600, 8), 4775, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(7²⁵, 2400, F₇, 8) (dual of [2400, 2375, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(7²⁵, 2401, F₇, 8) (dual of [2401, 2376, 9]-code), using
        an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(29−8, 29, 2545)-Net over F₇ — Digital

Digital (21, 29, 2545)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²⁹, 2545, F₇, 8) (dual of [2545, 2516, 9]-code), using
- 140 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 30 times 0, 1, 102 times 0) [i] based on linear OA(7²⁵, 2401, F₇, 8) (dual of [2401, 2376, 9]-code), using
  - an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(29−8, 29, 494151)-Net in Base 7 — Upper bound on s

There is no (21, 29, 494152)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 3 219909 062517 430070 615233 > 7²⁹ [i]