MinT - Best Known (43, 43+40, s)-Nets in Base 64

Best Known (43, 43+40, s)-Nets in Base 64

(43, 43+40, 513)-Net over F₆₄ — Constructive and digital

Digital (43, 83, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 83, 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]

(43, 43+40, 516)-Net in Base 64 — Constructive

(43, 83, 516)-net in base 64, using

1 times m-reduction [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)

(43, 43+40, 2055)-Net over F₆₄ — Digital

Digital (43, 83, 2055)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸³, 2055, F₆₄, 2, 40) (dual of [(2055, 2), 4027, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64⁸³, 4110, F₆₄, 40) (dual of [4110, 4027, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(34) [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(64⁶⁹, 4096, F₆₄, 35) (dual of [4096, 4027, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [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]

(43, 43+40, 4126782)-Net in Base 64 — Upper bound on s

There is no (43, 83, 4126783)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 818351 140673 465293 885803 228065 230969 915385 533664 033119 446935 290363 852050 400899 041003 151297 612407 541945 247751 734774 977230 348087 666279 116473 785702 679049 > 64⁸³ [i]