MinT - Best Known (143−15, 143, s)-Nets in Base 9

Best Known (143−15, 143, s)-Nets in Base 9

(143−15, 143, 2401127)-Net over F₉ — Constructive and digital

Digital (128, 143, 2401127)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (22, 29, 4385)-net over F₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 10)-net over F₉, using
      - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
        the rational function field F₉(x) [i]
        Niederreiter sequence [i]
    2. digital (19, 26, 4375)-net over F₉, using
      - net defined by OOA [i] based on linear OOA(9²⁶, 4375, F₉, 7, 7) (dual of [(4375, 7), 30599, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(9²⁶, 13126, F₉, 7) (dual of [13126, 13100, 8]-code), using
        trace code [i] based on linear OA(81¹³, 6563, F₈₁, 7) (dual of [6563, 6550, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(81¹³, 6561, F₈₁, 7) (dual of [6561, 6548, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(81¹¹, 6561, F₈₁, 6) (dual of [6561, 6550, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(81⁰, 2, F₈₁, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 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 (99, 114, 2396742)-net over F₉, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵⁷, 1198371, F₈₁, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(81⁵⁷, 8388598, F₈₁, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵⁷, large, F₈₁, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(143−15, 143, 2403303)-Net in Base 9 — Constructive

(128, 143, 2403303)-net in base 9, using

(u,Â u+v)-construction [i] based on
1. (22, 29, 6561)-net in base 9, using
  - net defined by OOA [i] based on OOA(9²⁹, 6561, S₉, 7, 7), using
    - OOA 3-folding and stacking with additional row [i] based on OA(9²⁹, 19684, S₉, 7), using
      - discarding factors based on OA(9²⁹, 19686, S₉, 7), using
        discarding parts of the base [i] based on linear OA(27¹⁹, 19686, F₂₇, 7) (dual of [19686, 19667, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(27¹⁹, 19683, F₂₇, 7) (dual of [19683, 19664, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(27¹⁶, 19683, F₂₇, 6) (dual of [19683, 19667, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(27⁰, 3, F₂₇, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 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 (99, 114, 2396742)-net over F₉, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵⁷, 1198371, F₈₁, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(81⁵⁷, 8388598, F₈₁, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵⁷, large, F₈₁, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(143−15, 143, large)-Net over F₉ — Digital

Digital (128, 143, large)-net over F₉, using

t-expansion [i] based on digital (124, 143, large)-net over F₉, using
- 2 times m-reduction [i] based on digital (124, 145, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(143−15, 143, large)-Net in Base 9 — Upper bound on s

There is no (128, 143, large)-net in base 9, because

13 times m-reduction [i] would yield (128, 130, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (143−15, 143, s)-Nets in Base 9

(143−15, 143, 2401127)-Net over F9 — Constructive and digital

(143−15, 143, 2403303)-Net in Base 9 — Constructive

(143−15, 143, large)-Net over F9 — Digital

(143−15, 143, large)-Net in Base 9 — Upper bound on s

(143−15, 143, 2401127)-Net over F₉ — Constructive and digital

(143−15, 143, large)-Net over F₉ — Digital