MinT - Best Known (141−21, 141, s)-Nets in Base 9

Best Known (141−21, 141, s)-Nets in Base 9

(141−21, 141, 478327)-Net over F₉ — Constructive and digital

Digital (120, 141, 478327)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (106, 127, 478297)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹²⁷, 478297, F₉, 21, 21) (dual of [(478297, 21), 10044110, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(9¹²⁷, 4782971, F₉, 21) (dual of [4782971, 4782844, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹²⁷, 4782976, F₉, 21) (dual of [4782976, 4782849, 22]-code), using
        constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        linear OA(9¹²⁷, 4782969, F₉, 21) (dual of [4782969, 4782842, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(9¹²⁰, 4782969, F₉, 20) (dual of [4782969, 4782849, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(9⁰, 7, F₉, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(141−21, 141, 5541492)-Net over F₉ — Digital

Digital (120, 141, 5541492)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(141−21, 141, large)-Net in Base 9 — Upper bound on s

There is no (120, 141, large)-net in base 9, because

19 times m-reduction [i] would yield (120, 122, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (141−21, 141, s)-Nets in Base 9

(141−21, 141, 478327)-Net over F9 — Constructive and digital

(141−21, 141, 5541492)-Net over F9 — Digital

(141−21, 141, large)-Net in Base 9 — Upper bound on s

(141−21, 141, 478327)-Net over F₉ — Constructive and digital

(141−21, 141, 5541492)-Net over F₉ — Digital