MinT - Best Known (50, 77, s)-Nets in Base 64

Best Known (50, 77, s)-Nets in Base 64

(50, 77, 715)-Net over F₆₄ — Constructive and digital

Digital (50, 77, 715)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 65)-net over F₆₄, using
  - kÂ =Â 2 construction [i]
2. digital (0, 2, 65)-net over F₆₄ (see above)
3. digital (0, 3, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
4. digital (0, 3, 65)-net over F₆₄ (see above)
5. digital (0, 3, 65)-net over F₆₄ (see above)
6. digital (0, 4, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
7. digital (0, 5, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
8. digital (0, 6, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
9. digital (0, 9, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
10. digital (0, 13, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
11. digital (0, 27, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(50, 77, 5042)-Net in Base 64 — Constructive

(50, 77, 5042)-net in base 64, using

64¹ times duplication [i] based on (49, 76, 5042)-net in base 64, using
- base change [i] based on digital (30, 57, 5042)-net over F₂₅₆, using
  - 256¹ times duplication [i] based on digital (29, 56, 5042)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁵⁶, 5042, F₂₅₆, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
      - OOA 13-folding and stacking with additional row [i] based on linear OA(256⁵⁶, 65547, F₂₅₆, 27) (dual of [65547, 65491, 28]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁵⁶, 65548, F₂₅₆, 27) (dual of [65548, 65492, 28]-code), using
        constructionÂ X applied to C([0,13]) âŠ‚ C([0,11]) [i] based on
        linear OA(256⁵³, 65537, F₂₅₆, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(256⁴⁵, 65537, F₂₅₆, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(50, 77, 37426)-Net over F₆₄ — Digital

Digital (50, 77, 37426)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 77, large)-Net in Base 64 — Upper bound on s

There is no (50, 77, large)-net in base 64, because

25 times m-reduction [i] would yield (50, 52, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (50, 77, s)-Nets in Base 64

(50, 77, 715)-Net over F64 — Constructive and digital

(50, 77, 5042)-Net in Base 64 — Constructive

(50, 77, 37426)-Net over F64 — Digital

(50, 77, large)-Net in Base 64 — Upper bound on s

(50, 77, 715)-Net over F₆₄ — Constructive and digital

(50, 77, 37426)-Net over F₆₄ — Digital