MinT - Best Known (24, 45, s)-Nets in Base 64

Best Known (24, 45, s)-Nets in Base 64

(24, 45, 410)-Net over F₆₄ — Constructive and digital

Digital (24, 45, 410)-net over F₆₄, using

64³ times duplication [i] based on 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
        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]
        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]
        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]

(24, 45, 516)-Net in Base 64 — Constructive

(24, 45, 516)-net in base 64, using

64¹ times duplication [i] based on (23, 44, 516)-net in base 64, using
- base change [i] based on digital (12, 33, 516)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 11, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (1, 22, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(24, 45, 2055)-Net over F₆₄ — Digital

Digital (24, 45, 2055)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁴⁵, 2055, F₆₄, 2, 21) (dual of [(2055, 2), 4065, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64⁴⁵, 4110, F₆₄, 21) (dual of [4110, 4065, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(15) [i] based on
    1. linear OA(64⁴¹, 4096, F₆₄, 21) (dual of [4096, 4055, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. linear OA(64³¹, 4096, F₆₄, 16) (dual of [4096, 4065, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(64⁴, 14, F₆₄, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
      - discarding factors / shortening the dual code based on linear OA(64⁴, 64, F₆₄, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
        Reedâ€“Solomon code RS(60,64) [i]

(24, 45, 6365423)-Net in Base 64 — Upper bound on s

There is no (24, 45, 6365424)-net in base 64, because

1 times m-reduction [i] would yield (24, 44, 6365424)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 29 642785 585386 348190 465450 573643 353988 212868 411957 674515 409347 988072 161144 393751 > 64⁴⁴ [i]