MinT - Best Known (78, 94, s)-Nets in Base 32

Best Known (78, 94, s)-Nets in Base 32

(78, 94, 1048834)-Net over F₃₂ — Constructive and digital

Digital (78, 94, 1048834)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (10, 18, 259)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32¹⁸, 259, F₃₂, 8, 8) (dual of [(259, 8), 2054, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(32¹⁸, 1036, F₃₂, 8) (dual of [1036, 1018, 9]-code), using
      - constructionÂ XX applied to C₁Â =Â C([1019,2]), C₂Â =Â C([0,3]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,2]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1019,3]) [i] based on
        linear OA(32¹³, 1023, F₃₂, 7) (dual of [1023, 1010, 8]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−4,−3,…,2}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(32⁷, 1023, F₃₂, 4) (dual of [1023, 1016, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32¹⁵, 1023, F₃₂, 8) (dual of [1023, 1008, 9]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−4,−3,…,3}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(32⁵, 1023, F₃₂, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32³, 11, F₃₂, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
        discarding factors / shortening the dual code based on linear OA(32³, 32, F₃₂, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
        Reedâ€“Solomon code RS(29,32) [i]
        linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 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 (60, 76, 1048575)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷⁶, 1048575, F₃₂, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(32⁷⁶, 8388600, F₃₂, 16) (dual of [8388600, 8388524, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁷⁶, large, F₃₂, 16) (dual of [large, large−76, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(78, 94, 1049631)-Net in Base 32 — Constructive

(78, 94, 1049631)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (12, 20, 1056)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 33)-net over F₃₂, using
      - s-reduction based on digital (0, 0, s)-net over F₃₂ with arbitrarily large s, using
        arbitrary set of 32⁰ points [i]
    2. digital (0, 0, 33)-net over F₃₂ (see above)
    3. digital (0, 0, 33)-net over F₃₂ (see above)
    4. digital (0, 0, 33)-net over F₃₂ (see above)
    5. digital (0, 0, 33)-net over F₃₂ (see above)
    6. digital (0, 0, 33)-net over F₃₂ (see above)
    7. digital (0, 0, 33)-net over F₃₂ (see above)
    8. digital (0, 0, 33)-net over F₃₂ (see above)
    9. digital (0, 0, 33)-net over F₃₂ (see above)
    10. digital (0, 0, 33)-net over F₃₂ (see above)
    11. digital (0, 0, 33)-net over F₃₂ (see above)
    12. digital (0, 0, 33)-net over F₃₂ (see above)
    13. digital (0, 0, 33)-net over F₃₂ (see above)
    14. digital (0, 0, 33)-net over F₃₂ (see above)
    15. digital (0, 0, 33)-net over F₃₂ (see above)
    16. digital (0, 0, 33)-net over F₃₂ (see above)
    17. digital (0, 0, 33)-net over F₃₂ (see above)
    18. digital (0, 0, 33)-net over F₃₂ (see above)
    19. digital (0, 0, 33)-net over F₃₂ (see above)
    20. digital (0, 0, 33)-net over F₃₂ (see above)
    21. digital (0, 0, 33)-net over F₃₂ (see above)
    22. digital (0, 0, 33)-net over F₃₂ (see above)
    23. digital (0, 0, 33)-net over F₃₂ (see above)
    24. digital (0, 0, 33)-net over F₃₂ (see above)
    25. digital (0, 1, 33)-net over F₃₂, using
      - s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    26. digital (0, 1, 33)-net over F₃₂ (see above)
    27. digital (0, 1, 33)-net over F₃₂ (see above)
    28. digital (0, 1, 33)-net over F₃₂ (see above)
    29. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    30. digital (0, 2, 33)-net over F₃₂ (see above)
    31. digital (0, 4, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
    32. digital (0, 8, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. (58, 74, 1048575)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁷⁴, 1048575, S₃₂, 16, 16), using
    - OA 8-folding and stacking [i] based on OA(32⁷⁴, 8388600, S₃₂, 16), using
      - discarding factors based on OA(32⁷⁴, large, S₃₂, 16), using
        discarding parts of the base [i] based on linear OA(64⁶¹, large, F₆₄, 16) (dual of [large, large−61, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(78, 94, large)-Net over F₃₂ — Digital

Digital (78, 94, large)-net over F₃₂, using

t-expansion [i] based on digital (75, 94, large)-net over F₃₂, using
- 1 times m-reduction [i] based on digital (75, 95, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(78, 94, large)-Net in Base 32 — Upper bound on s

There is no (78, 94, large)-net in base 32, because

14 times m-reduction [i] would yield (78, 80, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (78, 94, s)-Nets in Base 32

(78, 94, 1048834)-Net over F32 — Constructive and digital

(78, 94, 1049631)-Net in Base 32 — Constructive

(78, 94, large)-Net over F32 — Digital

(78, 94, large)-Net in Base 32 — Upper bound on s

(78, 94, 1048834)-Net over F₃₂ — Constructive and digital

(78, 94, large)-Net over F₃₂ — Digital