MinT - Best Known (46, 46+40, s)-Nets in Base 64

Best Known (46, 46+40, s)-Nets in Base 64

(46, 46+40, 513)-Net over F₆₄ — Constructive and digital

Digital (46, 86, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 86, 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]

(46, 46+40, 546)-Net in Base 64 — Constructive

(46, 86, 546)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (8, 28, 258)-net in base 64, using
  - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. (18, 58, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(46, 46+40, 2598)-Net over F₆₄ — Digital

Digital (46, 86, 2598)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁶, 2598, F₆₄, 40) (dual of [2598, 2512, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁸⁶, 4119, F₆₄, 40) (dual of [4119, 4033, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(31) [i] based on
    1. linear OA(64⁷⁹, 4096, F₆₄, 40) (dual of [4096, 4017, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(64⁶³, 4096, F₆₄, 32) (dual of [4096, 4033, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    3. linear OA(64⁷, 23, F₆₄, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,64)), using
      - discarding factors / shortening the dual code based on linear OA(64⁷, 64, F₆₄, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
        Reedâ€“Solomon code RS(57,64) [i]

(46, 46+40, 7700856)-Net in Base 64 — Upper bound on s

There is no (46, 86, 7700857)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 214525 352029 515750 306334 064202 395792 645180 449184 990111 221941 208756 994234 938107 985089 610110 316026 328866 656694 157014 095914 334040 359919 224948 244004 352887 738894 > 64⁸⁶ [i]