MinT - Best Known (142−15, 142, s)-Nets in Base 8

Best Known (142−15, 142, s)-Nets in Base 8

(142−15, 142, 2399476)-Net over F₈ — Constructive and digital

Digital (127, 142, 2399476)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (21, 28, 2734)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²⁸, 2734, F₈, 7, 7) (dual of [(2734, 7), 19110, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(8²⁸, 8203, F₈, 7) (dual of [8203, 8175, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(8²⁸, 8204, F₈, 7) (dual of [8204, 8176, 8]-code), using
        trace code [i] based on linear OA(64¹⁴, 4102, F₆₄, 7) (dual of [4102, 4088, 8]-code), using
        constructionÂ X applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(64¹³, 4097, F₆₄, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(64⁹, 4097, F₆₄, 5) (dual of [4097, 4088, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’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(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [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]

(142−15, 142, large)-Net over F₈ — Digital

Digital (127, 142, large)-net over F₈, using

t-expansion [i] based on digital (125, 142, large)-net over F₈, using
- 4 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]

(142−15, 142, large)-Net in Base 8 — Upper bound on s

There is no (127, 142, large)-net in base 8, because

13 times m-reduction [i] would yield (127, 129, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (142−15, 142, s)-Nets in Base 8

(142−15, 142, 2399476)-Net over F8 — Constructive and digital

(142−15, 142, large)-Net over F8 — Digital

(142−15, 142, large)-Net in Base 8 — Upper bound on s

(142−15, 142, 2399476)-Net over F₈ — Constructive and digital

(142−15, 142, large)-Net over F₈ — Digital