MinT - Best Known (91−16, 91, s)-Nets in Base 25

Best Known (91−16, 91, s)-Nets in Base 25

(91−16, 91, 1048732)-Net over F₂₅ — Constructive and digital

Digital (75, 91, 1048732)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 15, 157)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25¹⁵, 157, F₂₅, 8, 8) (dual of [(157, 8), 1241, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(25¹⁵, 628, F₂₅, 8) (dual of [628, 613, 9]-code), using
      - constructionÂ XX applied to C₁Â =Â C([623,5]), C₂Â =Â C([0,6]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,5]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([623,6]) [i] based on
        linear OA(25¹³, 624, F₂₅, 7) (dual of [624, 611, 8]-code), using the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {−1,0,…,5}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(25¹³, 624, F₂₅, 7) (dual of [624, 611, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(25¹⁵, 624, F₂₅, 8) (dual of [624, 609, 9]-code), using the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {−1,0,…,6}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(25¹¹, 624, F₂₅, 6) (dual of [624, 613, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(25⁰, 2, F₂₅, 0) (dual of [2, 2, 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]
        linear OA(25⁰, 2, F₂₅, 0) (dual of [2, 2, 1]-code) (see above)
2. digital (60, 76, 1048575)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁷⁶, 1048575, F₂₅, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(25⁷⁶, 8388600, F₂₅, 16) (dual of [8388600, 8388524, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁷⁶, large, F₂₅, 16) (dual of [large, large−76, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(91−16, 91, large)-Net over F₂₅ — Digital

Digital (75, 91, large)-net over F₂₅, using

t-expansion [i] based on digital (73, 91, large)-net over F₂₅, using
- 1 times m-reduction [i] based on digital (73, 92, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁹², large, F₂₅, 19) (dual of [large, large−92, 20]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25⁹¹, large, F₂₅, 19) (dual of [large, large−91, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(91−16, 91, large)-Net in Base 25 — Upper bound on s

There is no (75, 91, large)-net in base 25, because

14 times m-reduction [i] would yield (75, 77, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (91−16, 91, s)-Nets in Base 25

(91−16, 91, 1048732)-Net over F25 — Constructive and digital

(91−16, 91, large)-Net over F25 — Digital

(91−16, 91, large)-Net in Base 25 — Upper bound on s

(91−16, 91, 1048732)-Net over F₂₅ — Constructive and digital

(91−16, 91, large)-Net over F₂₅ — Digital