MinT - Best Known (42−21, 42, s)-Nets in Base 64

Best Known (42−21, 42, s)-Nets in Base 64

(42−21, 42, 410)-Net over F₆₄ — Constructive and digital

Digital (21, 42, 410)-net over F₆₄, using

net defined by OOA [i] based on linear OOA(64⁴², 410, F₆₄, 21, 21) (dual of [(410, 21), 8568, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(64⁴², 4101, F₆₄, 21) (dual of [4101, 4059, 22]-code), using
  - discarding factors / shortening the dual code based on linear OA(64⁴², 4102, F₆₄, 21) (dual of [4102, 4060, 22]-code), using
    - constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(64⁴¹, 4097, F₆₄, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      2. linear OA(64³⁷, 4097, F₆₄, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (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]

(42−21, 42, 514)-Net in Base 64 — Constructive

(21, 42, 514)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (4, 14, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
    - base change [i] based on digital (0, 12, 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. (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 (see above)

(42−21, 42, 1367)-Net over F₆₄ — Digital

Digital (21, 42, 1367)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁴², 1367, F₆₄, 3, 21) (dual of [(1367, 3), 4059, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(64⁴², 4101, F₆₄, 21) (dual of [4101, 4059, 22]-code), using
  - discarding factors / shortening the dual code based on linear OA(64⁴², 4102, F₆₄, 21) (dual of [4102, 4060, 22]-code), using
    - constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(64⁴¹, 4097, F₆₄, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      2. linear OA(64³⁷, 4097, F₆₄, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (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]

(42−21, 42, 1827984)-Net in Base 64 — Upper bound on s

There is no (21, 42, 1827985)-net in base 64, because

1 times m-reduction [i] would yield (21, 41, 1827985)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 113 078439 589640 969538 489553 903567 664491 452922 077844 739299 933488 481120 293640 > 64⁴¹ [i]