MinT - Best Known (100−15, 100, s)-Nets in Base 25

Best Known (100−15, 100, s)-Nets in Base 25

(100−15, 100, 1329206)-Net over F₂₅ — Constructive and digital

Digital (85, 100, 1329206)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (22, 29, 130835)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 626)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25⁴, 626, F₂₅, 3, 3) (dual of [(626, 3), 1874, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(25⁴, 626, F₂₅, 2, 3) (dual of [(626, 2), 1248, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁴, 626, F₂₅, 3) (dual of [626, 622, 4]-code or 626-cap in PG(3,25)), using
        ovoid in PG(3,Â 25) [i]
    2. digital (18, 25, 130209)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25²⁵, 130209, F₂₅, 7, 7) (dual of [(130209, 7), 911438, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(25²⁵, 390628, F₂₅, 7) (dual of [390628, 390603, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(25²⁵, 390629, F₂₅, 7) (dual of [390629, 390604, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(25²⁵, 390625, F₂₅, 7) (dual of [390625, 390600, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(25²¹, 390625, F₂₅, 6) (dual of [390625, 390604, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(25⁰, 4, F₂₅, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁰, s, F₂₅, 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 (56, 71, 1198371)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁷¹, 1198371, F₂₅, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(25⁷¹, 8388598, F₂₅, 15) (dual of [8388598, 8388527, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁷¹, large, F₂₅, 15) (dual of [large, large−71, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(100−15, 100, large)-Net over F₂₅ — Digital

Digital (85, 100, large)-net over F₂₅, using

7 times m-reduction [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]

(100−15, 100, large)-Net in Base 25 — Upper bound on s

There is no (85, 100, large)-net in base 25, because

13 times m-reduction [i] would yield (85, 87, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (100−15, 100, s)-Nets in Base 25

(100−15, 100, 1329206)-Net over F25 — Constructive and digital

(100−15, 100, large)-Net over F25 — Digital

(100−15, 100, large)-Net in Base 25 — Upper bound on s

(100−15, 100, 1329206)-Net over F₂₅ — Constructive and digital

(100−15, 100, large)-Net over F₂₅ — Digital