MinT - Best Known (111, 111+15, s)-Nets in Base 8

Best Known (111, 111+15, s)-Nets in Base 8

(111, 111+15, 2396770)-Net over F₈ — Constructive and digital

Digital (111, 126, 2396770)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 12, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (99, 114, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(111, 111+15, 2396775)-Net in Base 8 — Constructive

(111, 126, 2396775)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (5, 12, 33)-net in base 8, using
  - base change [i] based on digital (2, 9, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (99, 114, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(111, 111+15, large)-Net over F₈ — Digital

Digital (111, 126, large)-net over F₈, using

8² times duplication [i] based on digital (109, 124, large)-net over F₈, using
- t-expansion [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]

(111, 111+15, large)-Net in Base 8 — Upper bound on s

There is no (111, 126, large)-net in base 8, because

13 times m-reduction [i] would yield (111, 113, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]