MinT - Best Known (123−17, 123, s)-Nets in Base 8

Best Known (123−17, 123, s)-Nets in Base 8

(123−17, 123, 1048592)-Net over F₈ — Constructive and digital

Digital (106, 123, 1048592)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (96, 113, 1048575)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹¹³, 1048575, F₈, 17, 17) (dual of [(1048575, 17), 17825662, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(8¹¹³, 8388601, F₈, 17) (dual of [8388601, 8388488, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹¹³, large, F₈, 17) (dual of [large, large−113, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(123−17, 123, large)-Net over F₈ — Digital

Digital (106, 123, large)-net over F₈, using

1 times m-reduction [i] based on digital (106, 124, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²⁴, large, F₈, 18) (dual of [large, large−124, 19]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(8¹²¹, large, F₈, 18) (dual of [large, large−121, 19]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(123−17, 123, large)-Net in Base 8 — Upper bound on s

There is no (106, 123, large)-net in base 8, because

15 times m-reduction [i] would yield (106, 108, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]