MinT - Best Known (15, 15+7, s)-Nets in Base 7

Best Known (15, 15+7, s)-Nets in Base 7

(15, 15+7, 257)-Net over F₇ — Constructive and digital

Digital (15, 22, 257)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 3, 57)-net over F₇, using
  - kÂ =Â 2 construction [i]
2. digital (2, 5, 100)-net over F₇, using
  - s-reduction based on digital (2, 5, 132)-net over F₇, using
    - net defined by OOA [i] based on linear OOA(7⁵, 132, F₇, 3, 3) (dual of [(132, 3), 391, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(7⁵, 132, F₇, 2, 3) (dual of [(132, 2), 259, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7⁵, 132, F₇, 3) (dual of [132, 127, 4]-code or 132-cap in PG(4,7)), using
        caps completed using a computer search [i]
3. digital (7, 14, 100)-net over F₇, using
  - trace code 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]

(15, 15+7, 632)-Net over F₇ — Digital

Digital (15, 22, 632)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²², 632, F₇, 7) (dual of [632, 610, 8]-code), using
- 286 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 4 times 0, 1, 27 times 0, 1, 89 times 0, 1, 162 times 0) [i] based on linear OA(7¹⁸, 342, F₇, 7) (dual of [342, 324, 8]-code), using
  - 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]

(15, 15+7, 249411)-Net in Base 7 — Upper bound on s

There is no (15, 22, 249412)-net in base 7, because

1 times m-reduction [i] would yield (15, 21, 249412)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 558550 405233 216865 > 7²¹ [i]