MinT - Best Known (33−9, 33, s)-Nets in Base 8

Best Known (33−9, 33, s)-Nets in Base 8

(33−9, 33, 1032)-Net over F₈ — Constructive and digital

Digital (24, 33, 1032)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (20, 29, 1023)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²⁹, 1023, F₈, 9, 9) (dual of [(1023, 9), 9178, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(8²⁹, 4093, F₈, 9) (dual of [4093, 4064, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(8²⁹, 4096, F₈, 9) (dual of [4096, 4067, 10]-code), using
        an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(33−9, 33, 4249)-Net over F₈ — Digital

Digital (24, 33, 4249)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8³³, 4249, F₈, 9) (dual of [4249, 4216, 10]-code), using
- 149 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 32 times 0, 1, 108 times 0) [i] based on linear OA(8²⁹, 4096, F₈, 9) (dual of [4096, 4067, 10]-code), using
  - an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(33−9, 33, 5304866)-Net in Base 8 — Upper bound on s

There is no (24, 33, 5304867)-net in base 8, because

1 times m-reduction [i] would yield (24, 32, 5304867)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 79228 164902 019098 425948 792872 > 8³² [i]