MinT - Best Known (140−5, 140, s)-Nets in Base 8

Best Known (140−5, 140, s)-Nets in Base 8

(140−5, 140, large)-Net over F₈ — Constructive and digital

Digital (135, 140, large)-net over F₈, using

t-expansion [i] based on digital (133, 140, large)-net over F₈, using
- 4 times m-reduction [i] based on digital (133, 144, large)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (28, 33, 4194301)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8³³, 4194301, F₈, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(8³³, large, F₈, 5) (dual of [large, large−33, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’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, 4404019)-net over F₈, using
      - (u,Â u+v)-construction [i] based on
        digital (24, 29, 1048579)-net over F₈, using
        net defined by OOA [i] based on linear OOA(8²⁹, 1048579, F₈, 5, 5) (dual of [(1048579, 5), 5242866, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(8²⁹, 2097159, F₈, 5) (dual of [2097159, 2097130, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(8²⁹, 2097152, F₈, 5) (dual of [2097152, 2097123, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(8²², 2097152, F₈, 4) (dual of [2097152, 2097130, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(8⁰, 7, F₈, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
        digital (71, 82, 3355440)-net over F₈, using
        trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
        net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(140−5, 140, large)-Net in Base 8 — Upper bound on s

There is no (135, 140, large)-net in base 8, because

3 times m-reduction [i] would yield (135, 137, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (140−5, 140, s)-Nets in Base 8

(140−5, 140, large)-Net over F8 — Constructive and digital

(140−5, 140, large)-Net in Base 8 — Upper bound on s

(140−5, 140, large)-Net over F₈ — Constructive and digital