MinT - Best Known (151−14, 151, s)-Nets in Base 8

Best Known (151−14, 151, s)-Nets in Base 8

(151−14, 151, 3095798)-Net over F₈ — Constructive and digital

Digital (137, 151, 3095798)-net over F₈, using

8¹ times duplication [i] based on digital (136, 150, 3095798)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (37, 44, 699056)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁴⁴, 699056, F₈, 7, 7) (dual of [(699056, 7), 4893348, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(8⁴⁴, 2097169, F₈, 7) (dual of [2097169, 2097125, 8]-code), using
        constructionÂ X4 applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(8⁴³, 2097153, F₈, 7) (dual of [2097153, 2097110, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 8¹⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(8²⁹, 2097153, F₈, 5) (dual of [2097153, 2097124, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 8¹⁴âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(8¹⁵, 16, F₈, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,8)), using
        dual of repetition code with length 16 [i]
        linear OA(8¹, 16, F₈, 1) (dual of [16, 15, 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 (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]

(151−14, 151, large)-Net over F₈ — Digital

Digital (137, 151, large)-net over F₈, using

t-expansion [i] based on digital (131, 151, large)-net over F₈, using
- 2 times m-reduction [i] based on digital (131, 153, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵³, large, F₈, 22) (dual of [large, large−153, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(151−14, 151, large)-Net in Base 8 — Upper bound on s

There is no (137, 151, large)-net in base 8, because

12 times m-reduction [i] would yield (137, 139, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (151−14, 151, s)-Nets in Base 8

(151−14, 151, 3095798)-Net over F8 — Constructive and digital

(151−14, 151, large)-Net over F8 — Digital

(151−14, 151, large)-Net in Base 8 — Upper bound on s

(151−14, 151, 3095798)-Net over F₈ — Constructive and digital

(151−14, 151, large)-Net over F₈ — Digital