MinT - Best Known (28, 45, s)-Nets in Base 7

Best Known (28, 45, s)-Nets in Base 7

(28, 45, 115)-Net over F₇ — Constructive and digital

Digital (28, 45, 115)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (19, 36, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 18, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(28, 45, 314)-Net over F₇ — Digital

Digital (28, 45, 314)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁴⁵, 314, F₇, 17) (dual of [314, 269, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁴⁵, 342, F₇, 17) (dual of [342, 297, 18]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(28, 45, 27893)-Net in Base 7 — Upper bound on s

There is no (28, 45, 27894)-net in base 7, because

1 times m-reduction [i] would yield (28, 44, 27894)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 15 288954 319373 486634 111379 474909 952785 > 7⁴⁴ [i]