MinT - Best Known (52, 59, s)-Nets in Base 5

Best Known (52, 59, s)-Nets in Base 5

(52, 59, 2797915)-Net over F₅ — Constructive and digital

Digital (52, 59, 2797915)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁵⁹, 2797915, F₅, 9, 7) (dual of [(2797915, 9), 25181176, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(5⁵⁹, 2797916, F₅, 3, 7) (dual of [(2797916, 3), 8393689, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(5⁸, 1715, F₅, 3, 3) (dual of [(1715, 3), 5137, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(5⁸, 1715, F₅, 2, 3) (dual of [(1715, 2), 3422, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁸, 1715, F₅, 3) (dual of [1715, 1707, 4]-code or 1715-cap in PG(7,5)), using
        product of two caps with tangent hyperplane [i]
    2. linear OOA(5⁵¹, 2796201, F₅, 3, 7) (dual of [(2796201, 3), 8388552, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(5⁵¹, large, F₅, 7) (dual of [large, large−51, 8]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(52, 59, large)-Net over F₅ — Digital

Digital (52, 59, large)-net over F₅, using

5⁷ times duplication [i] based on digital (45, 52, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵², large, F₅, 7) (dual of [large, large−52, 8]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(5⁵¹, large, F₅, 7) (dual of [large, large−51, 8]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(52, 59, large)-Net in Base 5 — Upper bound on s

There is no (52, 59, large)-net in base 5, because

5 times m-reduction [i] would yield (52, 54, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]