MinT - Best Known (82, 82+11, s)-Nets in Base 25

Best Known (82, 82+11, s)-Nets in Base 25

(82, 82+11, 8388600)-Net over F₂₅ — Constructive and digital

Digital (82, 93, 8388600)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 6, 1677720)-net over F₂₅, using
  - s-reduction based on digital (4, 6, large)-net over F₂₅, using
    - kÂ =Â 2 construction [i]
2. digital (4, 6, 1677720)-net over F₂₅ (see above)
3. digital (6, 9, 1677720)-net over F₂₅, using
  - s-reduction based on digital (6, 9, large)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁹, large, F₂₅, 3, 3), using
      - appending kth column [i] based on linear OOA(25⁹, large, F₂₅, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁹, large, F₂₅, 3) (dual of [large, large−9, 4]-code), using
        product of projective cap with (hyper)oval [i] based on linear OA(25⁷, 391875, F₂₅, 3) (dual of [391875, 391868, 4]-code or 391875-cap in PG(6,25)), using
        product of two caps with tangent hyperplane [i]
4. digital (16, 21, 1677720)-net over F₂₅, using
  - s-reduction based on digital (16, 21, 4194301)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25²¹, 4194301, F₂₅, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(25²¹, large, F₂₅, 5) (dual of [large, large−21, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
5. digital (40, 51, 1677720)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁵¹, 1677720, F₂₅, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(25⁵¹, 8388601, F₂₅, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁵¹, large, F₂₅, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(82, 82+11, large)-Net over F₂₅ — Digital

Digital (82, 93, large)-net over F₂₅, using

t-expansion [i] based on digital (81, 93, large)-net over F₂₅, using
- 9 times m-reduction [i] based on digital (81, 102, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25¹⁰², large, F₂₅, 21) (dual of [large, large−102, 22]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25¹⁰¹, large, F₂₅, 21) (dual of [large, large−101, 22]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(82, 82+11, large)-Net in Base 25 — Upper bound on s

There is no (82, 93, large)-net in base 25, because

9 times m-reduction [i] would yield (82, 84, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (82, 82+11, s)-Nets in Base 25

(82, 82+11, 8388600)-Net over F25 — Constructive and digital

(82, 82+11, large)-Net over F25 — Digital

(82, 82+11, large)-Net in Base 25 — Upper bound on s

(82, 82+11, 8388600)-Net over F₂₅ — Constructive and digital

(82, 82+11, large)-Net over F₂₅ — Digital