MinT - Best Known (64−31, 64, s)-Nets in Base 64

Best Known (64−31, 64, s)-Nets in Base 64

(64−31, 64, 513)-Net over F₆₄ — Constructive and digital

Digital (33, 64, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 64, 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]

(64−31, 64, 516)-Net in Base 64 — Constructive

(33, 64, 516)-net in base 64, using

base change [i] based on digital (17, 48, 516)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 16, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
  2. digital (1, 32, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(64−31, 64, 1777)-Net over F₆₄ — Digital

Digital (33, 64, 1777)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁶⁴, 1777, F₆₄, 2, 31) (dual of [(1777, 2), 3490, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁶⁴, 2054, F₆₄, 2, 31) (dual of [(2054, 2), 4044, 32]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁶⁴, 4108, F₆₄, 31) (dual of [4108, 4044, 32]-code), using
    - constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
      1. linear OA(64⁶¹, 4097, F₆₄, 31) (dual of [4097, 4036, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
      2. 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]
      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]

(64−31, 64, 3929897)-Net in Base 64 — Upper bound on s

There is no (33, 64, 3929898)-net in base 64, because

1 times m-reduction [i] would yield (33, 63, 3929898)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 615658 395125 316149 702864 105248 546168 011217 417954 884211 139477 093394 652045 054727 271851 237568 008414 320675 267674 828976 > 64⁶³ [i]