MinT - Best Known (139−16, 139, s)-Nets in Base 8

Best Known (139−16, 139, s)-Nets in Base 8

(139−16, 139, 2097280)-Net over F₈ — Constructive and digital

Digital (123, 139, 2097280)-net over F₈, using

8¹ times duplication [i] based on digital (122, 138, 2097280)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 16, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 8, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (106, 122, 2097150)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8¹²², 2097150, F₈, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹²², 8388601, F₈, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(8¹²², 8388602, F₈, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
        trace code [i] based on linear OOA(64⁶¹, 4194301, F₆₄, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶¹, 8388602, F₆₄, 16) (dual of [8388602, 8388541, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶¹, large, F₆₄, 16) (dual of [large, large−61, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(139−16, 139, large)-Net over F₈ — Digital

Digital (123, 139, large)-net over F₈, using

t-expansion [i] based on digital (119, 139, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³⁹, large, F₈, 20) (dual of [large, large−139, 21]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(8¹³⁷, large, F₈, 20) (dual of [large, large−137, 21]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(139−16, 139, large)-Net in Base 8 — Upper bound on s

There is no (123, 139, large)-net in base 8, because

14 times m-reduction [i] would yield (123, 125, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (139−16, 139, s)-Nets in Base 8

(139−16, 139, 2097280)-Net over F8 — Constructive and digital

(139−16, 139, large)-Net over F8 — Digital

(139−16, 139, large)-Net in Base 8 — Upper bound on s

(139−16, 139, 2097280)-Net over F₈ — Constructive and digital

(139−16, 139, large)-Net over F₈ — Digital