MinT - Best Known (108−12, 108, s)-Nets in Base 32

Best Known (108−12, 108, s)-Nets in Base 32

(108−12, 108, 5593472)-Net over F₃₂ — Constructive and digital

Digital (96, 108, 5593472)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 174796)-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, 174796)-net over F₃₂ (see above)
3. digital (0, 0, 174796)-net over F₃₂ (see above)
4. digital (0, 0, 174796)-net over F₃₂ (see above)
5. digital (0, 0, 174796)-net over F₃₂ (see above)
6. digital (0, 0, 174796)-net over F₃₂ (see above)
7. digital (0, 0, 174796)-net over F₃₂ (see above)
8. digital (0, 0, 174796)-net over F₃₂ (see above)
9. digital (0, 0, 174796)-net over F₃₂ (see above)
10. digital (0, 0, 174796)-net over F₃₂ (see above)
11. digital (0, 0, 174796)-net over F₃₂ (see above)
12. digital (0, 0, 174796)-net over F₃₂ (see above)
13. digital (0, 0, 174796)-net over F₃₂ (see above)
14. digital (0, 0, 174796)-net over F₃₂ (see above)
15. digital (0, 0, 174796)-net over F₃₂ (see above)
16. digital (0, 0, 174796)-net over F₃₂ (see above)
17. digital (0, 0, 174796)-net over F₃₂ (see above)
18. digital (0, 0, 174796)-net over F₃₂ (see above)
19. digital (0, 0, 174796)-net over F₃₂ (see above)
20. digital (0, 0, 174796)-net over F₃₂ (see above)
21. digital (0, 1, 174796)-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]
22. digital (0, 1, 174796)-net over F₃₂ (see above)
23. digital (0, 1, 174796)-net over F₃₂ (see above)
24. digital (0, 1, 174796)-net over F₃₂ (see above)
25. digital (0, 1, 174796)-net over F₃₂ (see above)
26. digital (0, 1, 174796)-net over F₃₂ (see above)
27. digital (3, 5, 174796)-net over F₃₂, using
  - s-reduction based on digital (3, 5, 1082401)-net over F₃₂, using
    - kÂ =Â 2 construction [i]
28. digital (3, 5, 174796)-net over F₃₂ (see above)
29. digital (4, 7, 174796)-net over F₃₂, using
  - s-reduction based on digital (4, 7, 1050624)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32⁷, 1050624, F₃₂, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(32⁷, 1050624, F₃₂, 2, 3) (dual of [(1050624, 2), 2101241, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁷, 1050624, F₃₂, 3) (dual of [1050624, 1050617, 4]-code or 1050624-cap in PG(6,32)), using
        product of two caps with tangent hyperplane [i]
30. digital (9, 13, 174796)-net over F₃₂, using
  - s-reduction based on digital (9, 13, 524290)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32¹³, 524290, F₃₂, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(32¹³, 1048580, F₃₂, 4) (dual of [1048580, 1048567, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(32¹³, 1048576, F₃₂, 4) (dual of [1048576, 1048563, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁹, 1048576, F₃₂, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 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]
31. digital (15, 21, 174796)-net over F₃₂, using
  - s-reduction based on digital (15, 21, 349526)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32²¹, 349526, F₃₂, 6, 6) (dual of [(349526, 6), 2097135, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(32²¹, 1048578, F₃₂, 6) (dual of [1048578, 1048557, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(32²¹, 1048580, F₃₂, 6) (dual of [1048580, 1048559, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(32²¹, 1048576, F₃₂, 6) (dual of [1048576, 1048555, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(32¹⁷, 1048576, F₃₂, 5) (dual of [1048576, 1048559, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code) (see above)
32. digital (39, 51, 174796)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 6, 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]
    2. digital (33, 45, 174763)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32⁴⁵, 174763, F₃₂, 12, 12) (dual of [(174763, 12), 2097111, 13]-NRT-code), using
        OA 6-folding and stacking [i] based on linear OA(32⁴⁵, 1048578, F₃₂, 12) (dual of [1048578, 1048533, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁴⁵, 1048580, F₃₂, 12) (dual of [1048580, 1048535, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(32⁴⁵, 1048576, F₃₂, 12) (dual of [1048576, 1048531, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(32⁴¹, 1048576, F₃₂, 11) (dual of [1048576, 1048535, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code) (see above)

(108−12, 108, 5858705)-Net in Base 32 — Constructive

(96, 108, 5858705)-net in base 32, using

base change [i] based on digital (78, 90, 5858705)-net over F₆₄, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (2, 4, 266305)-net over F₆₄, using
    - kÂ =Â 2 construction [i]
  2. digital (4, 7, 1398100)-net over F₆₄, using
    - s-reduction based on digital (4, 7, large)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64⁷, large, F₆₄, 3, 3), using
        appending kth column [i] based on linear OOA(64⁷, large, F₆₄, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁷, large, F₆₄, 3) (dual of [large, large−7, 4]-code), using
        product of projective cap with 64²-ovoid cap in AG(3,Â 64) [i] based on linear OA(64⁴, 4097, F₆₄, 3) (dual of [4097, 4093, 4]-code or 4097-cap in PG(3,64)), using
        ovoid in PG(3,Â 64) [i]
  3. digital (9, 13, 1398100)-net over F₆₄, using
    - s-reduction based on digital (9, 13, 4194301)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64¹³, 4194301, F₆₄, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(64¹³, 4194301, F₆₄, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(64¹³, 8388602, F₆₄, 4) (dual of [8388602, 8388589, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹³, large, F₆₄, 4) (dual of [large, large−13, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
  4. digital (15, 21, 1398100)-net over F₆₄, using
    - s-reduction based on digital (15, 21, 2796201)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64²¹, 2796201, F₆₄, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  5. digital (33, 45, 1398100)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴⁵, 1398100, F₆₄, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
      - OA 6-folding and stacking [i] based on linear OA(64⁴⁵, 8388600, F₆₄, 12) (dual of [8388600, 8388555, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’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 F₃₂ — Digital

Digital (96, 108, large)-net over F₃₂, using

t-expansion [i] based on digital (87, 108, large)-net over F₃₂, using
- 2 times m-reduction [i] based on digital (87, 110, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(108−12, 108, large)-Net in Base 32 — Upper bound on s

There is no (96, 108, large)-net in base 32, because

10 times m-reduction [i] would yield (96, 98, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (108−12, 108, s)-Nets in Base 32

(108−12, 108, 5593472)-Net over F32 — Constructive and digital

(108−12, 108, 5858705)-Net in Base 32 — Constructive

(108−12, 108, large)-Net over F32 — Digital

(108−12, 108, large)-Net in Base 32 — Upper bound on s

(108−12, 108, 5593472)-Net over F₃₂ — Constructive and digital

(108−12, 108, large)-Net over F₃₂ — Digital