MinT - Best Known (150−42, 150, s)-Nets in Base 9

Best Known (150−42, 150, s)-Nets in Base 9

(150−42, 150, 804)-Net over F₉ — Constructive and digital

Digital (108, 150, 804)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (13, 34, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]
2. digital (74, 116, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 58, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(150−42, 150, 6570)-Net over F₉ — Digital

Digital (108, 150, 6570)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁵⁰, 6570, F₉, 42) (dual of [6570, 6420, 43]-code), using
- constructionÂ X applied to C_e(41) âŠ‚ C_e(39) [i] based on
  1. linear OA(9¹⁴⁹, 6561, F₉, 42) (dual of [6561, 6412, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
  2. linear OA(9¹⁴¹, 6561, F₉, 40) (dual of [6561, 6420, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
  3. linear OA(9¹, 9, F₉, 1) (dual of [9, 8, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]

(150−42, 150, 7102638)-Net in Base 9 — Upper bound on s

There is no (108, 150, 7102639)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 136891 537776 248431 342990 192057 369622 369076 944399 894011 595362 074179 889019 470328 131962 350318 879917 029891 421754 714292 275108 586408 113225 918745 233113 > 9¹⁵⁰ [i]