MinT - Best Known (29, 29+17, s)-Nets in Base 8

Best Known (29, 29+17, s)-Nets in Base 8

(29, 29+17, 256)-Net over F₈ — Constructive and digital

Digital (29, 46, 256)-net over F₈, using

2 times m-reduction [i] based on digital (29, 48, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 24, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(29, 29+17, 462)-Net over F₈ — Digital

Digital (29, 46, 462)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴⁶, 462, F₈, 17) (dual of [462, 416, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴⁶, 511, F₈, 17) (dual of [511, 465, 18]-code), using
  - the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−5,−4,…,11}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(29, 29+17, 514)-Net in Base 8 — Constructive

(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]

(29, 29+17, 64631)-Net in Base 8 — Upper bound on s

There is no (29, 46, 64632)-net in base 8, because

1 times m-reduction [i] would yield (29, 45, 64632)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 43560 627176 069278 659992 288836 592390 475670 > 8⁴⁵ [i]