MinT - Best Known (86−43, 86, s)-Nets in Base 64

Best Known (86−43, 86, s)-Nets in Base 64

(86−43, 86, 513)-Net over F₆₄ — Constructive and digital

Digital (43, 86, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 86, 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]

(86−43, 86, 514)-Net in Base 64 — Constructive

(43, 86, 514)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (7, 28, 257)-net in base 64, using
  - base change [i] based on digital (0, 21, 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. (15, 58, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (15, 60, 257)-net in base 64, using
    - base change [i] based on digital (0, 45, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(86−43, 86, 1534)-Net over F₆₄ — Digital

Digital (43, 86, 1534)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸⁶, 1534, F₆₄, 2, 43) (dual of [(1534, 2), 2982, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁸⁶, 2051, F₆₄, 2, 43) (dual of [(2051, 2), 4016, 44]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁸⁶, 4102, F₆₄, 43) (dual of [4102, 4016, 44]-code), using
    - constructionÂ X applied to C([0,21]) âŠ‚ C([0,20]) [i] based on
      1. linear OA(64⁸⁵, 4097, F₆₄, 43) (dual of [4097, 4012, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
      2. linear OA(64⁸¹, 4097, F₆₄, 41) (dual of [4097, 4016, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (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]

(86−43, 86, 2817597)-Net in Base 64 — Upper bound on s

There is no (43, 86, 2817598)-net in base 64, because

1 times m-reduction [i] would yield (43, 85, 2817598)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3351 974543 968286 212140 894698 931842 879258 369617 696645 237396 847776 516329 000927 685922 953438 094453 276536 156522 203325 708247 291904 107865 979976 810503 659743 349745 > 64⁸⁵ [i]