MinT - Best Known (47, 64, s)-Nets in Base 8

Best Known (47, 64, s)-Nets in Base 8

(47, 64, 515)-Net over F₈ — Constructive and digital

Digital (47, 64, 515)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁶⁴, 515, F₈, 17, 17) (dual of [(515, 17), 8691, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8⁶⁴, 4121, F₈, 17) (dual of [4121, 4057, 18]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁶⁴, 4123, F₈, 17) (dual of [4123, 4059, 18]-code), using
    - constructionÂ X applied to C_e(16) âŠ‚ C_e(10) [i] based on
      1. linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      2. linear OA(8³⁷, 4096, F₈, 11) (dual of [4096, 4059, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(8⁷, 27, F₈, 5) (dual of [27, 20, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
        a code by Danev and Olsson [i]

(47, 64, 674)-Net in Base 8 — Constructive

(47, 64, 674)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (10, 18, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 9, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. (29, 46, 514)-net in base 8, using
  - trace code for nets [i] based on (6, 23, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
      - base change [i] based on digital (0, 18, 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]

(47, 64, 4357)-Net over F₈ — Digital

Digital (47, 64, 4357)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁴, 4357, F₈, 17) (dual of [4357, 4293, 18]-code), using
- 250 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 7 times 0, 1, 22 times 0, 1, 61 times 0, 1, 154 times 0) [i] based on linear OA(8⁵⁸, 4101, F₈, 17) (dual of [4101, 4043, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(14) [i] based on
    1. linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(8⁵³, 4096, F₈, 15) (dual of [4096, 4043, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    3. linear OA(8¹, 5, F₈, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(47, 64, 6957054)-Net in Base 8 — Upper bound on s

There is no (47, 64, 6957055)-net in base 8, because

1 times m-reduction [i] would yield (47, 63, 6957055)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 784 638491 666499 263489 601912 607735 915380 801100 253541 517513 > 8⁶³ [i]