MinT - Best Known (49−20, 49, s)-Nets in Base 64

Best Known (49−20, 49, s)-Nets in Base 64

(49−20, 49, 513)-Net over F₆₄ — Constructive and digital

Digital (29, 49, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 49, 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]

(49−20, 49, 1639)-Net in Base 64 — Constructive

(29, 49, 1639)-net in base 64, using

base change [i] based on digital (22, 42, 1639)-net over F₁₂₈, using
- 1 times m-reduction [i] based on digital (22, 43, 1639)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴³, 1639, F₁₂₈, 21, 21) (dual of [(1639, 21), 34376, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(128⁴³, 16391, F₁₂₈, 21) (dual of [16391, 16348, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴³, 16392, F₁₂₈, 21) (dual of [16392, 16349, 22]-code), using
        constructionÂ X applied to C_e(20) âŠ‚ C_e(17) [i] based on
        linear OA(128⁴¹, 16384, F₁₂₈, 21) (dual of [16384, 16343, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(128³⁵, 16384, F₁₂₈, 18) (dual of [16384, 16349, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(128², 8, F₁₂₈, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
        discarding factors / shortening the dual code based on linear OA(128², 128, F₁₂₈, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
        Reedâ€“Solomon code RS(126,128) [i]

(49−20, 49, 5786)-Net over F₆₄ — Digital

Digital (29, 49, 5786)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴⁹, 5786, F₆₄, 20) (dual of [5786, 5737, 21]-code), using
- 1678 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 70 times 0, 1, 189 times 0, 1, 465 times 0, 1, 915 times 0) [i] based on linear OA(64³⁹, 4098, F₆₄, 20) (dual of [4098, 4059, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
    1. 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]
    2. linear OA(64³⁷, 4096, F₆₄, 19) (dual of [4096, 4059, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(49−20, 49, large)-Net in Base 64 — Upper bound on s

There is no (29, 49, large)-net in base 64, because

18 times m-reduction [i] would yield (29, 31, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (49−20, 49, s)-Nets in Base 64

(49−20, 49, 513)-Net over F64 — Constructive and digital

(49−20, 49, 1639)-Net in Base 64 — Constructive

(49−20, 49, 5786)-Net over F64 — Digital

(49−20, 49, large)-Net in Base 64 — Upper bound on s

(49−20, 49, 513)-Net over F₆₄ — Constructive and digital

(49−20, 49, 5786)-Net over F₆₄ — Digital