MinT - Best Known (47, 47+44, s)-Nets in Base 64

Best Known (47, 47+44, s)-Nets in Base 64

(47, 47+44, 513)-Net over F₆₄ — Constructive and digital

Digital (47, 91, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 91, 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]

(47, 47+44, 516)-Net in Base 64 — Constructive

(47, 91, 516)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (9, 31, 258)-net in base 64, using
  - 1 times m-reduction [i] based on (9, 32, 258)-net in base 64, using
    - base change [i] based on digital (1, 24, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. (16, 60, 258)-net in base 64, using
  - base change [i] based on digital (1, 45, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(47, 47+44, 2055)-Net over F₆₄ — Digital

Digital (47, 91, 2055)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁹¹, 2055, F₆₄, 2, 44) (dual of [(2055, 2), 4019, 45]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64⁹¹, 4110, F₆₄, 44) (dual of [4110, 4019, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(38) [i] based on
    1. linear OA(64⁸⁷, 4096, F₆₄, 44) (dual of [4096, 4009, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. linear OA(64⁷⁷, 4096, F₆₄, 39) (dual of [4096, 4019, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [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]

(47, 47+44, 4251366)-Net in Base 64 — Upper bound on s

There is no (47, 91, 4251367)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 230 345119 424959 805820 018101 921549 077030 908221 106566 519117 137418 376119 308369 797983 838439 512497 117697 481485 315272 497553 303514 887718 065245 260831 303326 935693 537338 862690 > 64⁹¹ [i]