MinT - Best Known (74−37, 74, s)-Nets in Base 64

Best Known (74−37, 74, s)-Nets in Base 64

(74−37, 74, 513)-Net over F₆₄ — Constructive and digital

Digital (37, 74, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 74, 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]

(74−37, 74, 514)-Net in Base 64 — Constructive

(37, 74, 514)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (6, 24, 257)-net in base 64, using
  - base change [i] based on digital (0, 18, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]
2. (13, 50, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (13, 52, 257)-net in base 64, using
    - base change [i] based on digital (0, 39, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(74−37, 74, 1419)-Net over F₆₄ — Digital

Digital (37, 74, 1419)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁷⁴, 1419, F₆₄, 2, 37) (dual of [(1419, 2), 2764, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁷⁴, 2051, F₆₄, 2, 37) (dual of [(2051, 2), 4028, 38]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁷⁴, 4102, F₆₄, 37) (dual of [4102, 4028, 38]-code), using
    - constructionÂ X applied to C([0,18]) âŠ‚ C([0,17]) [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(64⁶⁹, 4097, F₆₄, 35) (dual of [4097, 4028, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
      3. linear OA(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(74−37, 74, 2534260)-Net in Base 64 — Upper bound on s

There is no (37, 74, 2534261)-net in base 64, because

1 times m-reduction [i] would yield (37, 73, 2534261)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 709806 700711 907153 934248 118360 054700 265215 811899 715399 302392 269044 566724 147000 461192 536700 544646 628909 074736 794100 227761 355875 614490 > 64⁷³ [i]