MinT - Best Known (150, 150+16, s)-Nets in Base 8

Best Known (150, 150+16, s)-Nets in Base 8

(150, 150+16, 2228223)-Net over F₈ — Constructive and digital

Digital (150, 166, 2228223)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (36, 44, 131073)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁴⁴, 131073, F₈, 8, 8) (dual of [(131073, 8), 1048540, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(8⁴⁴, 524292, F₈, 8) (dual of [524292, 524248, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁴⁴, 524294, F₈, 8) (dual of [524294, 524250, 9]-code), using
        trace code [i] based on linear OA(64²², 262147, F₆₄, 8) (dual of [262147, 262125, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(64²², 262144, F₆₄, 8) (dual of [262144, 262122, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(64¹⁹, 262144, F₆₄, 7) (dual of [262144, 262125, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [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]

(150, 150+16, large)-Net over F₈ — Digital

Digital (150, 166, large)-net over F₈, using

t-expansion [i] based on digital (144, 166, large)-net over F₈, using
- 2 times m-reduction [i] based on digital (144, 168, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(150, 150+16, large)-Net in Base 8 — Upper bound on s

There is no (150, 166, large)-net in base 8, because

14 times m-reduction [i] would yield (150, 152, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (150, 150+16, s)-Nets in Base 8

(150, 150+16, 2228223)-Net over F8 — Constructive and digital

(150, 150+16, large)-Net over F8 — Digital

(150, 150+16, large)-Net in Base 8 — Upper bound on s

(150, 150+16, 2228223)-Net over F₈ — Constructive and digital

(150, 150+16, large)-Net over F₈ — Digital