MinT - Best Known (129, 144, s)-Nets in Base 9

Best Known (129, 144, s)-Nets in Base 9

(129, 144, 2401199)-Net over F₉ — Constructive and digital

Digital (129, 144, 2401199)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (23, 30, 4457)-net over F₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 82)-net over F₉, using
      - net defined by OOA [i] based on linear OOA(9⁴, 82, F₉, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(9⁴, 82, F₉, 2, 3) (dual of [(82, 2), 160, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9⁴, 82, F₉, 3) (dual of [82, 78, 4]-code or 82-cap in PG(3,9)), using
        ovoid in PG(3,Â 9) [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]

(129, 144, 2403305)-Net in Base 9 — Constructive

(129, 144, 2403305)-net in base 9, using

(u,Â u+v)-construction [i] based on
1. (23, 30, 6563)-net in base 9, using
  - base change [i] based on digital (13, 20, 6563)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27²⁰, 6563, F₂₇, 7, 7) (dual of [(6563, 7), 45921, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(27²⁰, 19690, F₂₇, 7) (dual of [19690, 19670, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(27²⁰, 19691, F₂₇, 7) (dual of [19691, 19671, 8]-code), using
        constructionÂ X applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(27¹⁹, 19684, F₂₇, 7) (dual of [19684, 19665, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(27¹³, 19684, F₂₇, 5) (dual of [19684, 19671, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(27¹, 7, F₂₇, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [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]

(129, 144, large)-Net over F₉ — Digital

Digital (129, 144, large)-net over F₉, using

t-expansion [i] based on digital (124, 144, large)-net over F₉, using
- 1 times m-reduction [i] based on digital (124, 145, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(129, 144, large)-Net in Base 9 — Upper bound on s

There is no (129, 144, large)-net in base 9, because

13 times m-reduction [i] would yield (129, 131, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (129, 144, s)-Nets in Base 9

(129, 144, 2401199)-Net over F9 — Constructive and digital

(129, 144, 2403305)-Net in Base 9 — Constructive

(129, 144, large)-Net over F9 — Digital

(129, 144, large)-Net in Base 9 — Upper bound on s

(129, 144, 2401199)-Net over F₉ — Constructive and digital

(129, 144, large)-Net over F₉ — Digital