MinT - Best Known (139−14, 139, s)-Nets in Base 8

Best Known (139−14, 139, s)-Nets in Base 8

(139−14, 139, 2407669)-Net over F₈ — Constructive and digital

Digital (125, 139, 2407669)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (26, 33, 10927)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8³³, 10927, F₈, 7, 7) (dual of [(10927, 7), 76456, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(8³³, 32782, F₈, 7) (dual of [32782, 32749, 8]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(8³², 32781, F₈, 7) (dual of [32781, 32749, 8]-code), using
        constructionÂ X4 applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(8³¹, 32769, F₈, 7) (dual of [32769, 32738, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 8¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(8²¹, 32769, F₈, 5) (dual of [32769, 32748, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 8¹⁰âˆ’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(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 (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]

(139−14, 139, large)-Net over F₈ — Digital

Digital (125, 139, large)-net over F₈, using

7 times m-reduction [i] based on digital (125, 146, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴⁶, large, F₈, 21) (dual of [large, large−146, 22]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8¹⁴⁵, large, F₈, 21) (dual of [large, large−145, 22]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(139−14, 139, large)-Net in Base 8 — Upper bound on s

There is no (125, 139, large)-net in base 8, because

12 times m-reduction [i] would yield (125, 127, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (139−14, 139, s)-Nets in Base 8

(139−14, 139, 2407669)-Net over F8 — Constructive and digital

(139−14, 139, large)-Net over F8 — Digital

(139−14, 139, large)-Net in Base 8 — Upper bound on s

(139−14, 139, 2407669)-Net over F₈ — Constructive and digital

(139−14, 139, large)-Net over F₈ — Digital