MinT - Best Known (56−15, 56, s)-Nets in Base 8

Best Known (56−15, 56, s)-Nets in Base 8

(56−15, 56, 587)-Net over F₈ — Constructive and digital

Digital (41, 56, 587)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁵⁶, 587, F₈, 15, 15) (dual of [(587, 15), 8749, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8⁵⁶, 4110, F₈, 15) (dual of [4110, 4054, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁵⁶, 4111, F₈, 15) (dual of [4111, 4055, 16]-code), using
    - constructionÂ X applied to C_e(14) âŠ‚ C_e(11) [i] based on
      1. 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]
      2. linear OA(8⁴¹, 4096, F₈, 12) (dual of [4096, 4055, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      3. linear OA(8³, 15, F₈, 2) (dual of [15, 12, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(8³, 63, F₈, 2) (dual of [63, 60, 3]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 8²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]

(56−15, 56, 674)-Net in Base 8 — Constructive

(41, 56, 674)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (9, 16, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 8, 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. (25, 40, 514)-net in base 8, using
  - base change [i] based on digital (15, 30, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 15, 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]

(56−15, 56, 4162)-Net over F₈ — Digital

Digital (41, 56, 4162)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁶, 4162, F₈, 15) (dual of [4162, 4106, 16]-code), using
- 59 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 51 times 0) [i] based on linear OA(8⁵³, 4100, F₈, 15) (dual of [4100, 4047, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
    1. 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]
    2. linear OA(8⁴⁹, 4096, F₈, 14) (dual of [4096, 4047, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(56−15, 56, 6019041)-Net in Base 8 — Upper bound on s

There is no (41, 56, 6019042)-net in base 8, because

1 times m-reduction [i] would yield (41, 55, 6019042)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 46 768062 715988 905638 922635 232466 777451 260146 335664 > 8⁵⁵ [i]