MinT - Best Known (56−27, 56, s)-Nets in Base 64

Best Known (56−27, 56, s)-Nets in Base 64

(56−27, 56, 513)-Net over F₆₄ — Constructive and digital

Digital (29, 56, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 56, 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]

(56−27, 56, 516)-Net in Base 64 — Constructive

(29, 56, 516)-net in base 64, using

base change [i] based on digital (15, 42, 516)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 14, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
  2. digital (1, 28, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(56−27, 56, 1791)-Net over F₆₄ — Digital

Digital (29, 56, 1791)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁵⁶, 1791, F₆₄, 2, 27) (dual of [(1791, 2), 3526, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁵⁶, 2054, F₆₄, 2, 27) (dual of [(2054, 2), 4052, 28]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁵⁶, 4108, F₆₄, 27) (dual of [4108, 4052, 28]-code), using
    - constructionÂ X applied to C([0,13]) âŠ‚ C([0,11]) [i] based on
      1. linear OA(64⁵³, 4097, F₆₄, 27) (dual of [4097, 4044, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
      2. linear OA(64⁴⁵, 4097, F₆₄, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
      3. linear OA(64³, 11, F₆₄, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
        discarding factors / shortening the dual code based on linear OA(64³, 64, F₆₄, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
        Reedâ€“Solomon code RS(61,64) [i]

(56−27, 56, 3940901)-Net in Base 64 — Upper bound on s

There is no (29, 56, 3940902)-net in base 64, because

1 times m-reduction [i] would yield (29, 55, 3940902)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2187 252772 824668 446799 971996 160763 922145 043959 845646 237816 936677 131984 065615 482938 599053 980337 168348 > 64⁵⁵ [i]