MinT - Best Known (27, 43, s)-Nets in Base 7

Best Known (27, 43, s)-Nets in Base 7

(27, 43, 115)-Net over F₇ — Constructive and digital

Digital (27, 43, 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 (18, 34, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 17, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(27, 43, 338)-Net over F₇ — Digital

Digital (27, 43, 338)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁴³, 338, F₇, 16) (dual of [338, 295, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁴³, 342, F₇, 16) (dual of [342, 299, 17]-code), using
  - the primitive BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â {−5,−4,…,10}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(27, 43, 21869)-Net in Base 7 — Upper bound on s

There is no (27, 43, 21870)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2 183973 896344 491985 690385 276124 027025 > 7⁴³ [i]