MinT - Best Known (140, 140+17, s)-Nets in Base 8

Best Known (140, 140+17, s)-Nets in Base 8

(140, 140+17, 2097483)-Net over F₈ — Constructive and digital

Digital (140, 157, 2097483)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (19, 27, 333)-net over F₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (1, 3, 73)-net over F₈, using
      - kÂ =Â 2 construction [i]
    2. digital (4, 8, 130)-net over F₈, using
      - trace code for nets [i] based on digital (0, 4, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
    3. digital (8, 16, 130)-net over F₈, using
      - trace code for nets [i] based on digital (0, 8, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
2. digital (113, 130, 2097150)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³⁰, 2097150, F₈, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹³⁰, 8388601, F₈, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹³⁰, 8388602, F₈, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
        trace code [i] based on linear OOA(64⁶⁵, 4194301, F₆₄, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶⁵, 8388602, F₆₄, 17) (dual of [8388602, 8388537, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(140, 140+17, 2097678)-Net in Base 8 — Constructive

(140, 157, 2097678)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (19, 27, 528)-net in base 8, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 5, 14)-net over F₈, using
      - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
        F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
    2. (14, 22, 514)-net in base 8, using
      - trace code for nets [i] based on (3, 11, 257)-net in base 64, using
        1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
        base change [i] based on digital (0, 9, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. digital (113, 130, 2097150)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³⁰, 2097150, F₈, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹³⁰, 8388601, F₈, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹³⁰, 8388602, F₈, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
        trace code [i] based on linear OOA(64⁶⁵, 4194301, F₆₄, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶⁵, 8388602, F₆₄, 17) (dual of [8388602, 8388537, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(140, 140+17, large)-Net over F₈ — Digital

Digital (140, 157, large)-net over F₈, using

t-expansion [i] based on digital (138, 157, large)-net over F₈, using
- 4 times m-reduction [i] based on digital (138, 161, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(140, 140+17, large)-Net in Base 8 — Upper bound on s

There is no (140, 157, large)-net in base 8, because

15 times m-reduction [i] would yield (140, 142, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (140, 140+17, s)-Nets in Base 8

(140, 140+17, 2097483)-Net over F8 — Constructive and digital

(140, 140+17, 2097678)-Net in Base 8 — Constructive

(140, 140+17, large)-Net over F8 — Digital

(140, 140+17, large)-Net in Base 8 — Upper bound on s

(140, 140+17, 2097483)-Net over F₈ — Constructive and digital

(140, 140+17, large)-Net over F₈ — Digital