MinT - Best Known (172−14, 172, s)-Nets in Base 8

Best Known (172−14, 172, s)-Nets in Base 8

(172−14, 172, 4809874)-Net over F₈ — Constructive and digital

Digital (158, 172, 4809874)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (13, 17, 16390)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁷, 16390, F₈, 4, 4) (dual of [(16390, 4), 65543, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹⁷, 16390, F₈, 3, 4) (dual of [(16390, 3), 49153, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8¹⁷, 32780, F₈, 4) (dual of [32780, 32763, 5]-code), using
        constructionÂ X4 applied to C_e(3) âŠ‚ C_e(1) [i] based on
        linear OA(8¹⁶, 32768, F₈, 4) (dual of [32768, 32752, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(8⁶, 32768, F₈, 2) (dual of [32768, 32762, 3]-code), using an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
        linear OA(8¹¹, 12, F₈, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,8)), using
        dual of repetition code with length 12 [i]
        linear OA(8¹, 12, F₈, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹, 511, F₈, 1) (dual of [511, 510, 2]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,0], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
2. digital (42, 49, 2396742)-net over F₈, using
  - s-reduction based on digital (42, 49, 2796200)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁴⁹, 2796200, F₈, 7, 7) (dual of [(2796200, 7), 19573351, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(8⁴⁹, 8388601, F₈, 7) (dual of [8388601, 8388552, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁴⁹, large, F₈, 7) (dual of [large, large−49, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
3. digital (92, 106, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(172−14, 172, 4858764)-Net in Base 8 — Constructive

(158, 172, 4858764)-net in base 8, using

base change [i] based on digital (115, 129, 4858764)-net over F₁₆, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (8, 12, 65280)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16¹², 65280, F₁₆, 4, 4) (dual of [(65280, 4), 261108, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(16¹², 130560, F₁₆, 4) (dual of [130560, 130548, 5]-code), using
        trace code [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
  2. digital (30, 37, 2396742)-net over F₁₆, using
    - s-reduction based on digital (30, 37, 2796200)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16³⁷, 2796200, F₁₆, 7, 7) (dual of [(2796200, 7), 19573363, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(16³⁷, 8388601, F₁₆, 7) (dual of [8388601, 8388564, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(16³⁷, large, F₁₆, 7) (dual of [large, large−37, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
  3. digital (66, 80, 2396742)-net over F₁₆, using
    - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
        OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(172−14, 172, large)-Net over F₈ — Digital

Digital (158, 172, large)-net over F₈, using

8⁴ times duplication [i] based on digital (154, 168, large)-net over F₈, using
- t-expansion [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]

(172−14, 172, large)-Net in Base 8 — Upper bound on s

There is no (158, 172, large)-net in base 8, because

12 times m-reduction [i] would yield (158, 160, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (172−14, 172, s)-Nets in Base 8

(172−14, 172, 4809874)-Net over F8 — Constructive and digital

(172−14, 172, 4858764)-Net in Base 8 — Constructive

(172−14, 172, large)-Net over F8 — Digital

(172−14, 172, large)-Net in Base 8 — Upper bound on s

(172−14, 172, 4809874)-Net over F₈ — Constructive and digital

(172−14, 172, large)-Net over F₈ — Digital