MinT - Best Known (131−36, 131, s)-Nets in Base 9

Best Known (131−36, 131, s)-Nets in Base 9

(131−36, 131, 780)-Net over F₉ — Constructive and digital

Digital (95, 131, 780)-net over F₉, using

1 times m-reduction [i] based on digital (95, 132, 780)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 26, 40)-net over F₉, using
    - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
        a function field by SÃ©mirat [i]
  2. digital (69, 106, 740)-net over F₉, using
    - trace code for nets [i] based on digital (16, 53, 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]

(131−36, 131, 6572)-Net over F₉ — Digital

Digital (95, 131, 6572)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³¹, 6572, F₉, 36) (dual of [6572, 6441, 37]-code), using
- constructionÂ X applied to C([0,18]) âŠ‚ C([0,16]) [i] based on
  1. linear OA(9¹²⁹, 6562, F₉, 37) (dual of [6562, 6433, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 9⁸âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
  2. linear OA(9¹²¹, 6562, F₉, 33) (dual of [6562, 6441, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 9⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
  3. linear OA(9², 10, F₉, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,9)), using
    - extended Reedâ€“Solomon code RS_e(8,9) [i]
    - Hamming code H(2,9) [i]

(131−36, 131, 8313893)-Net in Base 9 — Upper bound on s

There is no (95, 131, 8313894)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 101337 351268 110180 996721 040177 369433 899442 133790 845975 623866 036791 821892 056468 681774 466169 924328 098529 893533 009771 193121 410209 > 9¹³¹ [i]