MinT - Best Known (39, 53, s)-Nets in Base 8

Best Known (39, 53, s)-Nets in Base 8

(39, 53, 588)-Net over F₈ — Constructive and digital

Digital (39, 53, 588)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁵³, 588, F₈, 14, 14) (dual of [(588, 14), 8179, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(8⁵³, 4116, F₈, 14) (dual of [4116, 4063, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(9) [i] based on
    1. 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]
    2. linear OA(8³³, 4096, F₈, 10) (dual of [4096, 4063, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    3. linear OA(8⁴, 20, F₈, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,8)), using
      - discarding factors / shortening the dual code based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]

(39, 53, 644)-Net in Base 8 — Constructive

(39, 53, 644)-net in base 8, using

1 times m-reduction [i] based on (39, 54, 644)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 14, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 7, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [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]

(39, 53, 4334)-Net over F₈ — Digital

Digital (39, 53, 4334)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵³, 4334, F₈, 14) (dual of [4334, 4281, 15]-code), using
- 230 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 46 times 0, 1, 175 times 0) [i] based on linear OA(8⁴⁹, 4100, F₈, 14) (dual of [4100, 4051, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
    1. 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]
    2. linear OA(8⁴⁵, 4096, F₈, 13) (dual of [4096, 4051, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [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]

(39, 53, 3322778)-Net in Base 8 — Upper bound on s

There is no (39, 53, 3322779)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 730751 310208 938926 166878 570358 860649 367392 025736 > 8⁵³ [i]