MinT - Best Known (114−10, 114, s)-Nets in Base 9

Best Known (114−10, 114, s)-Nets in Base 9

(114−10, 114, 7158757)-Net over F₉ — Constructive and digital

Digital (104, 114, 7158757)-net over F₉, using

generalized (u,Â u+v)-construction [i] based on
1. 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]
2. 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]
3. digital (64, 74, 3355440)-net over F₉, using
  - trace code for nets [i] based on digital (27, 37, 1677720)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81³⁷, 1677720, F₈₁, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(81³⁷, 8388600, F₈₁, 10) (dual of [8388600, 8388563, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(81³⁷, large, F₈₁, 10) (dual of [large, large−37, 11]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(114−10, 114, large)-Net over F₉ — Digital

Digital (104, 114, large)-net over F₉, using

t-expansion [i] based on digital (101, 114, large)-net over F₉, using
- 4 times m-reduction [i] based on digital (101, 118, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(114−10, 114, large)-Net in Base 9 — Upper bound on s

There is no (104, 114, large)-net in base 9, because

8 times m-reduction [i] would yield (104, 106, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (114−10, 114, s)-Nets in Base 9

(114−10, 114, 7158757)-Net over F9 — Constructive and digital

(114−10, 114, large)-Net over F9 — Digital

(114−10, 114, large)-Net in Base 9 — Upper bound on s

(114−10, 114, 7158757)-Net over F₉ — Constructive and digital

(114−10, 114, large)-Net over F₉ — Digital