MinT - Best Known (85−41, 85, s)-Nets in Base 64

Best Known (85−41, 85, s)-Nets in Base 64

(85−41, 85, 513)-Net over F₆₄ — Constructive and digital

Digital (44, 85, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 85, 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]

(85−41, 85, 516)-Net in Base 64 — Constructive

(44, 85, 516)-net in base 64, using

64¹ times duplication [i] based on (43, 84, 516)-net in base 64, using
- base change [i] based on digital (22, 63, 516)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. 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. digital (1, 42, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(85−41, 85, 2055)-Net over F₆₄ — Digital

Digital (44, 85, 2055)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸⁵, 2055, F₆₄, 2, 41) (dual of [(2055, 2), 4025, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64⁸⁵, 4110, F₆₄, 41) (dual of [4110, 4025, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(35) [i] based on
    1. linear OA(64⁸¹, 4096, F₆₄, 41) (dual of [4096, 4015, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(64⁷¹, 4096, F₆₄, 36) (dual of [4096, 4025, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    3. linear OA(64⁴, 14, F₆₄, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
      - discarding factors / shortening the dual code based on linear OA(64⁴, 64, F₆₄, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
        Reedâ€“Solomon code RS(60,64) [i]

(85−41, 85, 5080667)-Net in Base 64 — Upper bound on s

There is no (44, 85, 5080668)-net in base 64, because

1 times m-reduction [i] would yield (44, 84, 5080668)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 52 374451 399275 314160 292715 101339 713917 161266 928798 997649 122271 912860 012068 713582 407790 815124 507736 256096 802629 921886 983414 186950 057810 574813 659050 918934 > 64⁸⁴ [i]