MinT - Best Known (173−14, 173, s)-Nets in Base 8

Best Known (173−14, 173, s)-Nets in Base 8

(173−14, 173, 4809879)-Net over F₈ — Constructive and digital

Digital (159, 173, 4809879)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (14, 18, 16395)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁸, 16395, F₈, 4, 4) (dual of [(16395, 4), 65562, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹⁸, 16395, F₈, 3, 4) (dual of [(16395, 3), 49167, 5]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(8², 9, F₈, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;25,8) [i]
        linear OOA(8¹⁶, 16386, F₈, 3, 4) (dual of [(16386, 3), 49142, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(8¹⁶, 32772, F₈, 4) (dual of [32772, 32756, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹⁶, 32773, F₈, 4) (dual of [32773, 32757, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [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(8¹¹, 32768, F₈, 3) (dual of [32768, 32757, 4]-code or 32768-cap in PG(10,8)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 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 (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]

(173−14, 173, 4858764)-Net in Base 8 — Constructive

(159, 173, 4858764)-net in base 8, using

8¹ times duplication [i] based on (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]

(173−14, 173, large)-Net over F₈ — Digital

Digital (159, 173, large)-net over F₈, using

8⁵ 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]

(173−14, 173, large)-Net in Base 8 — Upper bound on s

There is no (159, 173, large)-net in base 8, because

12 times m-reduction [i] would yield (159, 161, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (173−14, 173, s)-Nets in Base 8

(173−14, 173, 4809879)-Net over F8 — Constructive and digital

(173−14, 173, 4858764)-Net in Base 8 — Constructive

(173−14, 173, large)-Net over F8 — Digital

(173−14, 173, large)-Net in Base 8 — Upper bound on s

(173−14, 173, 4809879)-Net over F₈ — Constructive and digital

(173−14, 173, large)-Net over F₈ — Digital