MinT - Best Known (66−31, 66, s)-Nets in Base 64

Best Known (66−31, 66, s)-Nets in Base 64

(66−31, 66, 513)-Net over F₆₄ — Constructive and digital

Digital (35, 66, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 66, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(66−31, 66, 517)-Net in Base 64 — Constructive

(35, 66, 517)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (7, 22, 258)-net in base 64, using
  - 2 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
    - base change [i] based on digital (1, 18, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. (13, 44, 259)-net in base 64, using
  - base change [i] based on digital (2, 33, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(66−31, 66, 2057)-Net over F₆₄ — Digital

Digital (35, 66, 2057)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁶⁶, 2057, F₆₄, 31) (dual of [2057, 1991, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁶⁶, 4114, F₆₄, 31) (dual of [4114, 4048, 32]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(64⁶¹, 4097, F₆₄, 31) (dual of [4097, 4036, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(64⁴⁹, 4097, F₆₄, 25) (dual of [4097, 4048, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(64⁵, 17, F₆₄, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
      - discarding factors / shortening the dual code based on linear OA(64⁵, 64, F₆₄, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
        Reedâ€“Solomon code RS(59,64) [i]

(66−31, 66, 6842353)-Net in Base 64 — Upper bound on s

There is no (35, 66, 6842354)-net in base 64, because

1 times m-reduction [i] would yield (35, 65, 6842354)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2521 730156 177206 759395 817229 814505 889826 377144 448457 386142 479118 257752 145385 207969 401676 078343 430257 436173 315721 868264 > 64⁶⁵ [i]