MinT - Best Known (40, 40+37, s)-Nets in Base 64

Best Known (40, 40+37, s)-Nets in Base 64

(40, 40+37, 513)-Net over F₆₄ — Constructive and digital

Digital (40, 77, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 77, 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]

(40, 40+37, 516)-Net in Base 64 — Constructive

(40, 77, 516)-net in base 64, using

64¹ times duplication [i] based on (39, 76, 516)-net in base 64, using
- base change [i] based on digital (20, 57, 516)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 19, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (1, 38, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(40, 40+37, 2055)-Net over F₆₄ — Digital

Digital (40, 77, 2055)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁷⁷, 2055, F₆₄, 2, 37) (dual of [(2055, 2), 4033, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64⁷⁷, 4110, F₆₄, 37) (dual of [4110, 4033, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(31) [i] based on
    1. linear OA(64⁷³, 4096, F₆₄, 37) (dual of [4096, 4023, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [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(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]

(40, 40+37, 5068529)-Net in Base 64 — Upper bound on s

There is no (40, 77, 5068530)-net in base 64, because

1 times m-reduction [i] would yield (40, 76, 5068530)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 186071 048558 758114 336207 926071 764969 304876 800870 456476 463219 817610 685572 091322 338587 735841 607205 099954 828354 975241 101022 067893 833321 192784 > 64⁷⁶ [i]