MinT - Best Known (76−37, 76, s)-Nets in Base 64

Best Known (76−37, 76, s)-Nets in Base 64

(76−37, 76, 513)-Net over F₆₄ — Constructive and digital

Digital (39, 76, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 76, 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]

(76−37, 76, 516)-Net in Base 64 — Constructive

(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)

(76−37, 76, 1817)-Net over F₆₄ — Digital

Digital (39, 76, 1817)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁷⁶, 1817, F₆₄, 2, 37) (dual of [(1817, 2), 3558, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁷⁶, 2054, F₆₄, 2, 37) (dual of [(2054, 2), 4032, 38]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁷⁶, 4108, F₆₄, 37) (dual of [4108, 4032, 38]-code), using
    - constructionÂ X applied to C([0,18]) âŠ‚ C([0,16]) [i] based on
      1. linear OA(64⁷³, 4097, F₆₄, 37) (dual of [4097, 4024, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
      2. linear OA(64⁶⁵, 4097, F₆₄, 33) (dual of [4097, 4032, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (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]

(76−37, 76, 4022892)-Net in Base 64 — Upper bound on s

There is no (39, 76, 4022893)-net in base 64, because

1 times m-reduction [i] would yield (39, 75, 4022893)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2907 358644 509425 407228 382308 452829 790610 864832 609624 438146 353684 736947 519158 824925 044471 853284 006127 070347 620985 660640 093317 564580 644460 > 64⁷⁵ [i]