MinT - Best Known (14, 22, s)-Nets in Base 7

Best Known (14, 22, s)-Nets in Base 7

(14, 22, 121)-Net over F₇ — Constructive and digital

Digital (14, 22, 121)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 21)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁶, 21, F₇, 4, 4) (dual of [(21, 4), 78, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(7⁶, 21, F₇, 3, 4) (dual of [(21, 3), 57, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(7⁶, 42, F₇, 4) (dual of [42, 36, 5]-code), using
        1 times truncation [i] based on linear OA(7⁷, 43, F₇, 5) (dual of [43, 36, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (8, 16, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 8, 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]

(14, 22, 365)-Net over F₇ — Digital

Digital (14, 22, 365)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²², 365, F₇, 8) (dual of [365, 343, 9]-code), using
- 19 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 14 times 0) [i] based on linear OA(7¹⁹, 343, F₇, 8) (dual of [343, 324, 9]-code), using
  - an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(14, 22, 16401)-Net in Base 7 — Upper bound on s

There is no (14, 22, 16402)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 3 910303 265019 524233 > 7²² [i]