MinT - Best Known (123, 123+15, s)-Nets in Base 9

Best Known (123, 123+15, s)-Nets in Base 9

(123, 123+15, 2397480)-Net over F₉ — Constructive and digital

Digital (123, 138, 2397480)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (17, 24, 738)-net over F₉, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 82)-net over F₉, using
      - s-reduction based on digital (0, 0, s)-net over F₉ with arbitrarily large s, using
        arbitrary set of 9⁰ points [i]
    2. digital (0, 0, 82)-net over F₉ (see above)
    3. digital (0, 1, 82)-net over F₉, using
      - s-reduction based on digital (0, 1, s)-net over F₉ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    4. digital (0, 1, 82)-net over F₉ (see above)
    5. digital (0, 1, 82)-net over F₉ (see above)
    6. digital (0, 1, 82)-net over F₉ (see above)
    7. digital (1, 3, 82)-net over F₉, using
      - s-reduction based on digital (1, 3, 91)-net over F₉, using
        kÂ =Â 2 construction [i]
    8. digital (1, 4, 82)-net over F₉, using
      - net defined by OOA [i] based on linear OOA(9⁴, 82, F₉, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(9⁴, 82, F₉, 2, 3) (dual of [(82, 2), 160, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9⁴, 82, F₉, 3) (dual of [82, 78, 4]-code or 82-cap in PG(3,9)), using
        ovoid in PG(3,Â 9) [i]
    9. digital (6, 13, 82)-net over F₉, using
      - base reduction for projective spaces (embedding PG(6,81) in PG(12,9)) for nets [i] based on digital (0, 7, 82)-net over F₈₁, using
        net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
2. digital (99, 114, 2396742)-net over F₉, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵⁷, 1198371, F₈₁, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(81⁵⁷, 8388598, F₈₁, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵⁷, large, F₈₁, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(123, 123+15, 2397498)-Net in Base 9 — Constructive

(123, 138, 2397498)-net in base 9, using

(u,Â u+v)-construction [i] based on
1. (17, 24, 756)-net in base 9, using
  - base change [i] based on digital (9, 16, 756)-net over F₂₇, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 0, 28)-net over F₂₇, using
        s-reduction based on digital (0, 0, s)-net over F₂₇ with arbitrarily large s, using
        arbitrary set of 27⁰ points [i]
      2. digital (0, 0, 28)-net over F₂₇ (see above)
      3. digital (0, 0, 28)-net over F₂₇ (see above)
      4. digital (0, 0, 28)-net over F₂₇ (see above)
      5. digital (0, 0, 28)-net over F₂₇ (see above)
      6. digital (0, 0, 28)-net over F₂₇ (see above)
      7. digital (0, 0, 28)-net over F₂₇ (see above)
      8. digital (0, 0, 28)-net over F₂₇ (see above)
      9. digital (0, 0, 28)-net over F₂₇ (see above)
      10. digital (0, 0, 28)-net over F₂₇ (see above)
      11. digital (0, 0, 28)-net over F₂₇ (see above)
      12. digital (0, 0, 28)-net over F₂₇ (see above)
      13. digital (0, 0, 28)-net over F₂₇ (see above)
      14. digital (0, 0, 28)-net over F₂₇ (see above)
      15. digital (0, 0, 28)-net over F₂₇ (see above)
      16. digital (0, 0, 28)-net over F₂₇ (see above)
      17. digital (0, 0, 28)-net over F₂₇ (see above)
      18. digital (0, 0, 28)-net over F₂₇ (see above)
      19. digital (0, 0, 28)-net over F₂₇ (see above)
      20. digital (0, 0, 28)-net over F₂₇ (see above)
      21. digital (0, 1, 28)-net over F₂₇, using
        s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
      22. digital (0, 1, 28)-net over F₂₇ (see above)
      23. digital (0, 1, 28)-net over F₂₇ (see above)
      24. digital (0, 1, 28)-net over F₂₇ (see above)
      25. digital (0, 2, 28)-net over F₂₇, using
        kÂ =Â 2 construction [i]
      26. digital (0, 3, 28)-net over F₂₇, using
        net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]
      27. digital (0, 7, 28)-net over F₂₇, using
        net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (99, 114, 2396742)-net over F₉, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵⁷, 1198371, F₈₁, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(81⁵⁷, 8388598, F₈₁, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵⁷, large, F₈₁, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(123, 123+15, large)-Net over F₉ — Digital

Digital (123, 138, large)-net over F₉, using

9¹ times duplication [i] based on digital (122, 137, large)-net over F₉, using
- t-expansion [i] based on digital (117, 137, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³⁷, large, F₉, 20) (dual of [large, large−137, 21]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(123, 123+15, large)-Net in Base 9 — Upper bound on s

There is no (123, 138, large)-net in base 9, because

13 times m-reduction [i] would yield (123, 125, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (123, 123+15, s)-Nets in Base 9

(123, 123+15, 2397480)-Net over F9 — Constructive and digital

(123, 123+15, 2397498)-Net in Base 9 — Constructive

(123, 123+15, large)-Net over F9 — Digital

(123, 123+15, large)-Net in Base 9 — Upper bound on s

(123, 123+15, 2397480)-Net over F₉ — Constructive and digital

(123, 123+15, large)-Net over F₉ — Digital