MinT - Best Known (35, 53, s)-Nets in Base 8

Best Known (35, 53, s)-Nets in Base 8

(35, 53, 354)-Net over F₈ — Constructive and digital

Digital (35, 53, 354)-net over F₈, using

3 times m-reduction [i] based on digital (35, 56, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 28, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(35, 53, 516)-Net in Base 8 — Constructive

(35, 53, 516)-net in base 8, using

1 times m-reduction [i] based on (35, 54, 516)-net in base 8, using
- trace code for nets [i] based on (8, 27, 258)-net in base 64, using
  - 1 times m-reduction [i] based on (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]

(35, 53, 681)-Net over F₈ — Digital

Digital (35, 53, 681)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵³, 681, F₈, 18) (dual of [681, 628, 19]-code), using
- 159 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 0, 1, 10 times 0, 1, 25 times 0, 1, 47 times 0, 1, 68 times 0) [i] based on linear OA(8⁴⁶, 515, F₈, 18) (dual of [515, 469, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
    1. linear OA(8⁴⁶, 512, F₈, 18) (dual of [512, 466, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(8⁴³, 512, F₈, 17) (dual of [512, 469, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 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]

(35, 53, 123262)-Net in Base 8 — Upper bound on s

There is no (35, 53, 123263)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 730771 382408 860091 650937 743534 606844 798655 334178 > 8⁵³ [i]