MinT - Best Known (108−12, 108, s)-Nets in Base 25

Best Known (108−12, 108, s)-Nets in Base 25

(108−12, 108, 5592400)-Net over F₂₅ — Constructive and digital

Digital (96, 108, 5592400)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (20, 26, 2796201)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25²⁶, 2796201, F₂₅, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(25²⁶, large, F₂₅, 6) (dual of [large, large−26, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
2. digital (70, 82, 2796200)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁸², 2796200, F₂₅, 14, 12) (dual of [(2796200, 14), 39146718, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(25⁸², 8388601, F₂₅, 2, 12) (dual of [(8388601, 2), 16777120, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(25⁸², 8388602, F₂₅, 2, 12) (dual of [(8388602, 2), 16777122, 13]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(25²⁶, 4194301, F₂₅, 2, 6) (dual of [(4194301, 2), 8388576, 7]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25²⁶, 8388602, F₂₅, 6) (dual of [8388602, 8388576, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(25²⁶, large, F₂₅, 6) (dual of [large, large−26, 7]-code) (see above)
        linear OOA(25⁵⁶, 4194301, F₂₅, 2, 12) (dual of [(4194301, 2), 8388546, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25⁵⁶, 8388602, F₂₅, 12) (dual of [8388602, 8388546, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁵⁶, large, F₂₅, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(108−12, 108, large)-Net over F₂₅ — Digital

Digital (96, 108, large)-net over F₂₅, using

25¹ times duplication [i] based on digital (95, 107, large)-net over F₂₅, using
- t-expansion [i] based on digital (85, 107, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25¹⁰⁷, large, F₂₅, 22) (dual of [large, large−107, 23]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25¹⁰⁶, large, F₂₅, 22) (dual of [large, large−106, 23]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(108−12, 108, large)-Net in Base 25 — Upper bound on s

There is no (96, 108, large)-net in base 25, because

10 times m-reduction [i] would yield (96, 98, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (108−12, 108, s)-Nets in Base 25

(108−12, 108, 5592400)-Net over F25 — Constructive and digital

(108−12, 108, large)-Net over F25 — Digital

(108−12, 108, large)-Net in Base 25 — Upper bound on s

(108−12, 108, 5592400)-Net over F₂₅ — Constructive and digital

(108−12, 108, large)-Net over F₂₅ — Digital