MinT - Best Known (72−13, 72, s)-Nets in Base 9

Best Known (72−13, 72, s)-Nets in Base 9

(72−13, 72, 88577)-Net over F₉ — Constructive and digital

Digital (59, 72, 88577)-net over F₉, using

9¹ times duplication [i] based on digital (58, 71, 88577)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁷¹, 88577, F₉, 13, 13) (dual of [(88577, 13), 1151430, 14]-NRT-code), using
  - OOA 6-folding and stacking with additional row [i] based on linear OA(9⁷¹, 531463, F₉, 13) (dual of [531463, 531392, 14]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(9⁷⁰, 531462, F₉, 13) (dual of [531462, 531392, 14]-code), using
      - constructionÂ X applied to C_e(12) âŠ‚ C_e(9) [i] based on
        linear OA(9⁶⁷, 531441, F₉, 13) (dual of [531441, 531374, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(9⁴⁹, 531441, F₉, 10) (dual of [531441, 531392, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(9³, 21, F₉, 2) (dual of [21, 18, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(9³, 80, F₉, 2) (dual of [80, 77, 3]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 9²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]

(72−13, 72, 531466)-Net over F₉ — Digital

Digital (59, 72, 531466)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁷², 531466, F₉, 13) (dual of [531466, 531394, 14]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9⁷⁰, 531462, F₉, 13) (dual of [531462, 531392, 14]-code), using
    - constructionÂ X applied to C_e(12) âŠ‚ C_e(9) [i] based on
      1. linear OA(9⁶⁷, 531441, F₉, 13) (dual of [531441, 531374, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      2. linear OA(9⁴⁹, 531441, F₉, 10) (dual of [531441, 531392, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
      3. linear OA(9³, 21, F₉, 2) (dual of [21, 18, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(9³, 80, F₉, 2) (dual of [80, 77, 3]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 9²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
  2. linear OA(9⁷⁰, 531464, F₉, 12) (dual of [531464, 531394, 13]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9⁷⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 205586 077138 049805 229100 691626 015791 635194 115897 049997 412325 874553 [i]
  3. linear OA(9⁰, 2, F₉, 0) (dual of [2, 2, 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]

(72−13, 72, large)-Net in Base 9 — Upper bound on s

There is no (59, 72, large)-net in base 9, because

11 times m-reduction [i] would yield (59, 61, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (72−13, 72, s)-Nets in Base 9

(72−13, 72, 88577)-Net over F9 — Constructive and digital

(72−13, 72, 531466)-Net over F9 — Digital

(72−13, 72, large)-Net in Base 9 — Upper bound on s

(72−13, 72, 88577)-Net over F₉ — Constructive and digital

(72−13, 72, 531466)-Net over F₉ — Digital