MinT - Best Known (120−12, 120, s)-Nets in Base 9

Best Known (120−12, 120, s)-Nets in Base 9

(120−12, 120, 2815966)-Net over F₉ — Constructive and digital

Digital (108, 120, 2815966)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (24, 30, 19766)-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 (20, 26, 19684)-net over F₉, using
      - net defined by OOA [i] based on linear OOA(9²⁶, 19684, F₉, 6, 6) (dual of [(19684, 6), 118078, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(9²⁶, 59052, F₉, 6) (dual of [59052, 59026, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(9²⁶, 59054, F₉, 6) (dual of [59054, 59028, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(9²⁶, 59049, F₉, 6) (dual of [59049, 59023, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(9²¹, 59049, F₉, 5) (dual of [59049, 59028, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(9⁰, 5, F₉, 0) (dual of [5, 5, 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 (78, 90, 2796200)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁹⁰, 2796200, F₉, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁰, 8388601, F₉, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9⁹⁰, 8388602, F₉, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁵, 4194301, F₈₁, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁴⁵, 8388602, F₈₁, 12) (dual of [8388602, 8388557, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁵, large, F₈₁, 12) (dual of [large, large−45, 13]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(120−12, 120, large)-Net over F₉ — Digital

Digital (108, 120, large)-net over F₉, using

t-expansion [i] based on digital (107, 120, large)-net over F₉, using
- 5 times m-reduction [i] based on digital (107, 125, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(120−12, 120, large)-Net in Base 9 — Upper bound on s

There is no (108, 120, large)-net in base 9, because

10 times m-reduction [i] would yield (108, 110, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (120−12, 120, s)-Nets in Base 9

(120−12, 120, 2815966)-Net over F9 — Constructive and digital

(120−12, 120, large)-Net over F9 — Digital

(120−12, 120, large)-Net in Base 9 — Upper bound on s

(120−12, 120, 2815966)-Net over F₉ — Constructive and digital

(120−12, 120, large)-Net over F₉ — Digital