MinT - Best Known (137−4, 137, s)-Nets in Base 9

Best Known (137−4, 137, s)-Nets in Base 9

(137−4, 137, large)-Net over F₉ — Constructive and digital

Digital (133, 137, large)-net over F₉, using

7 times m-reduction [i] based on digital (133, 144, large)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (28, 33, 4194301)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9³³, 4194301, F₉, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(9³³, large, F₉, 5) (dual of [large, large−33, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 9¹⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
  2. digital (100, 111, 5746927)-net over F₉, using
    - (u,Â u+v)-construction [i] based on
      1. digital (24, 29, 2391487)-net over F₉, using
        net defined by OOA [i] based on linear OOA(9²⁹, 2391487, F₉, 5, 5) (dual of [(2391487, 5), 11957406, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(9²⁹, 4782975, F₉, 5) (dual of [4782975, 4782946, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(9²⁹, 4782976, F₉, 5) (dual of [4782976, 4782947, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(9²⁹, 4782969, F₉, 5) (dual of [4782969, 4782940, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(9²², 4782969, F₉, 4) (dual of [4782969, 4782947, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [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]
      2. digital (71, 82, 3355440)-net over F₉, using
        trace code for nets [i] based on digital (30, 41, 1677720)-net over F₈₁, using
        net defined by OOA [i] based on linear OOA(81⁴¹, 1677720, F₈₁, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(81⁴¹, 8388601, F₈₁, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴¹, large, F₈₁, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(137−4, 137, large)-Net in Base 9 — Upper bound on s

There is no (133, 137, large)-net in base 9, because

2 times m-reduction [i] would yield (133, 135, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (137−4, 137, s)-Nets in Base 9

(137−4, 137, large)-Net over F9 — Constructive and digital

(137−4, 137, large)-Net in Base 9 — Upper bound on s

(137−4, 137, large)-Net over F₉ — Constructive and digital