MinT - Best Known (31−6, 31, s)-Nets in Base 25

Best Known (31−6, 31, s)-Nets in Base 25

(31−6, 31, 2797735)-Net over F₂₅ — Constructive and digital

Digital (25, 31, 2797735)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 5, 1534)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁵, 1534, F₂₅, 3, 3) (dual of [(1534, 3), 4597, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(25⁵, 1534, F₂₅, 2, 3) (dual of [(1534, 2), 3063, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁵, 1534, F₂₅, 3) (dual of [1534, 1529, 4]-code or 1534-cap in PG(4,25)), using
        large caps in PG(4,Â b) [i]
2. 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]

(31−6, 31, large)-Net over F₂₅ — Digital

Digital (25, 31, large)-net over F₂₅, using

t-expansion [i] based on digital (24, 31, large)-net over F₂₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25³¹, large, F₂₅, 7) (dual of [large, large−31, 8]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(31−6, 31, large)-Net in Base 25 — Upper bound on s

There is no (25, 31, large)-net in base 25, because

4 times m-reduction [i] would yield (25, 27, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]