MinT - Best Known (111, 111+4, s)-Nets in Base 9

Best Known (111, 111+4, s)-Nets in Base 9

(111, 111+4, large)-Net over F₉ — Constructive and digital

Digital (111, 115, large)-net over F₉, using

t-expansion [i] based on digital (107, 115, large)-net over F₉, using
- 2 times m-reduction [i] based on digital (107, 117, large)-net over F₉, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 1, 956593)-net over F₉, using
      - s-reduction based on digital (0, 1, s)-net over F₉ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    2. digital (0, 1, 956593)-net over F₉ (see above)
    3. digital (0, 1, 956593)-net over F₉ (see above)
    4. digital (0, 1, 956593)-net over F₉ (see above)
    5. digital (6, 8, 956593)-net over F₉, using
      - s-reduction based on digital (6, 8, 5380840)-net over F₉, using
        kÂ =Â 2 construction [i]
    6. digital (6, 8, 956593)-net over F₉ (see above)
    7. digital (8, 11, 956593)-net over F₉, using
      - s-reduction based on digital (8, 11, 1411830)-net over F₉, using
        net defined by OOA [i] based on linear OOA(9¹¹, 1411830, F₉, 3, 3) (dual of [(1411830, 3), 4235479, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(9¹¹, 1411830, F₉, 2, 3) (dual of [(1411830, 2), 2823649, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9¹¹, 1411830, F₉, 3) (dual of [1411830, 1411819, 4]-code or 1411830-cap in PG(10,9)), using
        product of projective cap with cap in AG(4,Â 9) [i] based on linear OA(9⁷, 6723, F₉, 3) (dual of [6723, 6716, 4]-code or 6723-cap in PG(6,9)), using
        product of two caps with tangent hyperplane [i]
    8. digital (24, 29, 956593)-net over F₉, using
      - s-reduction based on digital (24, 29, 2391487)-net over F₉, using
        net defined by OOA [i] based on linear OOA(9²⁹, 2391487, F₉, 5, 5) (dual of [(2391487, 5), 11957406, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(9²⁹, 4782975, F₉, 5) (dual of [4782975, 4782946, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(9²⁹, 4782976, F₉, 5) (dual of [4782976, 4782947, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(9²⁹, 4782969, F₉, 5) (dual of [4782969, 4782940, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(9²², 4782969, F₉, 4) (dual of [4782969, 4782947, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(9⁰, 7, F₉, 0) (dual of [7, 7, 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]
    9. digital (47, 57, 956593)-net over F₉, using
      - net defined by OOA [i] based on linear OOA(9⁵⁷, 956593, F₉, 10, 10) (dual of [(956593, 10), 9565873, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(9⁵⁷, 4782965, F₉, 10) (dual of [4782965, 4782908, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁵⁷, 4782969, F₉, 10) (dual of [4782969, 4782912, 11]-code), using
        an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(111, 111+4, large)-Net in Base 9 — Upper bound on s

There is no (111, 115, large)-net in base 9, because

2 times m-reduction [i] would yield (111, 113, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (111, 111+4, s)-Nets in Base 9

(111, 111+4, large)-Net over F9 — Constructive and digital

(111, 111+4, large)-Net in Base 9 — Upper bound on s

(111, 111+4, large)-Net over F₉ — Constructive and digital