MinT - Best Known (141−15, 141, s)-Nets in Base 8

Best Known (141−15, 141, s)-Nets in Base 8

(141−15, 141, 2399474)-Net over F₈ — Constructive and digital

Digital (126, 141, 2399474)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (20, 27, 2732)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²⁷, 2732, F₈, 7, 7) (dual of [(2732, 7), 19097, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(8²⁷, 8197, F₈, 7) (dual of [8197, 8170, 8]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(8²⁶, 8196, F₈, 7) (dual of [8196, 8170, 8]-code), using
        trace code [i] based on linear OA(64¹³, 4098, F₆₄, 7) (dual of [4098, 4085, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(64¹³, 4096, F₆₄, 7) (dual of [4096, 4083, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(64¹¹, 4096, F₆₄, 6) (dual of [4096, 4085, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 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 (99, 114, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(141−15, 141, large)-Net over F₈ — Digital

Digital (126, 141, large)-net over F₈, using

t-expansion [i] based on digital (125, 141, large)-net over F₈, using
- 5 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]

(141−15, 141, large)-Net in Base 8 — Upper bound on s

There is no (126, 141, large)-net in base 8, because

13 times m-reduction [i] would yield (126, 128, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (141−15, 141, s)-Nets in Base 8

(141−15, 141, 2399474)-Net over F8 — Constructive and digital

(141−15, 141, large)-Net over F8 — Digital

(141−15, 141, large)-Net in Base 8 — Upper bound on s

(141−15, 141, 2399474)-Net over F₈ — Constructive and digital

(141−15, 141, large)-Net over F₈ — Digital