MinT - Best Known (42−20, 42, s)-Nets in Base 64

Best Known (42−20, 42, s)-Nets in Base 64

(42−20, 42, 410)-Net over F₆₄ — Constructive and digital

Digital (22, 42, 410)-net over F₆₄, using

t-expansion [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]

(42−20, 42, 515)-Net in Base 64 — Constructive

(22, 42, 515)-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. (8, 28, 258)-net in base 64, using
  - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(42−20, 42, 2017)-Net over F₆₄ — Digital

Digital (22, 42, 2017)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁴², 2017, F₆₄, 2, 20) (dual of [(2017, 2), 3992, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁴², 2053, F₆₄, 2, 20) (dual of [(2053, 2), 4064, 21]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁴², 4106, F₆₄, 20) (dual of [4106, 4064, 21]-code), using
    - discarding factors / shortening the dual code based on linear OA(64⁴², 4107, F₆₄, 20) (dual of [4107, 4065, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(15) [i] based on
        linear OA(64³⁹, 4096, F₆₄, 20) (dual of [4096, 4057, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        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]
        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]

(42−20, 42, 2770708)-Net in Base 64 — Upper bound on s

There is no (22, 42, 2770709)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 7237 005757 091166 537132 078512 920515 111794 714101 565355 866678 113579 551273 496180 > 64⁴² [i]

Best Known (42−20, 42, s)-Nets in Base 64

(42−20, 42, 410)-Net over F64 — Constructive and digital

(42−20, 42, 515)-Net in Base 64 — Constructive

(42−20, 42, 2017)-Net over F64 — Digital

(42−20, 42, 2770708)-Net in Base 64 — Upper bound on s

(42−20, 42, 410)-Net over F₆₄ — Constructive and digital

(42−20, 42, 2017)-Net over F₆₄ — Digital