MinT - Best Known (98−9, 98, s)-Nets in Base 5

Best Known (98−9, 98, s)-Nets in Base 5

(98−9, 98, 4195865)-Net over F₅ — Constructive and digital

Digital (89, 98, 4195865)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (12, 16, 1565)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹⁶, 1565, F₅, 4, 4) (dual of [(1565, 4), 6244, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(5¹⁶, 1565, F₅, 3, 4) (dual of [(1565, 3), 4679, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(5¹⁶, 3130, F₅, 4) (dual of [3130, 3114, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(5¹⁶, 3125, F₅, 4) (dual of [3125, 3109, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(5¹¹, 3125, F₅, 3) (dual of [3125, 3114, 4]-code or 3125-cap in PG(10,5)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(5⁰, 5, F₅, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 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 (73, 82, 4194300)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁸², 4194300, F₅, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(5⁸², 8388601, F₅, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5⁸², 8388602, F₅, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
        trace code [i] based on linear OOA(25⁴¹, 4194301, F₂₅, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25⁴¹, 8388602, F₂₅, 9) (dual of [8388602, 8388561, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁴¹, large, F₂₅, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(98−9, 98, large)-Net over F₅ — Digital

Digital (89, 98, large)-net over F₅, using

t-expansion [i] based on digital (88, 98, large)-net over F₅, using
- 2 times m-reduction [i] based on digital (88, 100, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁰, large, F₅, 12) (dual of [large, large−100, 13]-code), using
    - 9 times code embedding in larger space [i] based on linear OA(5⁹¹, large, F₅, 12) (dual of [large, large−91, 13]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(98−9, 98, large)-Net in Base 5 — Upper bound on s

There is no (89, 98, large)-net in base 5, because

7 times m-reduction [i] would yield (89, 91, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (98−9, 98, s)-Nets in Base 5

(98−9, 98, 4195865)-Net over F5 — Constructive and digital

(98−9, 98, large)-Net over F5 — Digital

(98−9, 98, large)-Net in Base 5 — Upper bound on s

(98−9, 98, 4195865)-Net over F₅ — Constructive and digital

(98−9, 98, large)-Net over F₅ — Digital