MinT - Best Known (36, 36+21, s)-Nets in Base 8

Best Known (36, 36+21, s)-Nets in Base 8

(36, 36+21, 354)-Net over F₈ — Constructive and digital

Digital (36, 57, 354)-net over F₈, using

1 times m-reduction [i] based on digital (36, 58, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 29, 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]

(36, 36+21, 509)-Net over F₈ — Digital

Digital (36, 57, 509)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁷, 509, F₈, 21) (dual of [509, 452, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵⁷, 511, F₈, 21) (dual of [511, 454, 22]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(36, 36+21, 514)-Net in Base 8 — Constructive

(36, 57, 514)-net in base 8, using

8¹ times duplication [i] based on (35, 56, 514)-net in base 8, using
- base change [i] based on digital (21, 42, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 21, 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]

(36, 36+21, 73815)-Net in Base 8 — Upper bound on s

There is no (36, 57, 73816)-net in base 8, because

1 times m-reduction [i] would yield (36, 56, 73816)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 374 171948 977115 817647 256867 995371 240190 631742 738922 > 8⁵⁶ [i]