MinT - Best Known (139, 139+15, s)-Nets in Base 8

Best Known (139, 139+15, s)-Nets in Base 8

(139, 139+15, 2571510)-Net over F₈ — Constructive and digital

Digital (139, 154, 2571510)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (33, 40, 174768)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁴⁰, 174768, F₈, 7, 7) (dual of [(174768, 7), 1223336, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(8⁴⁰, 524305, F₈, 7) (dual of [524305, 524265, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁴⁰, 524306, F₈, 7) (dual of [524306, 524266, 8]-code), using
        trace code [i] based on linear OA(64²⁰, 262153, F₆₄, 7) (dual of [262153, 262133, 8]-code), using
        constructionÂ X4 applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(64¹⁹, 262145, F₆₄, 7) (dual of [262145, 262126, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(64¹³, 262145, F₆₄, 5) (dual of [262145, 262132, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’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(64⁷, 8, F₆₄, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,64)), using
        dual of repetition code with length 8 [i]
        linear OA(64¹, 8, F₆₄, 1) (dual of [8, 7, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, 64, F₆₄, 1) (dual of [64, 63, 2]-code), using
        Reedâ€“Solomon code RS(63,64) [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]

(139, 139+15, large)-Net over F₈ — Digital

Digital (139, 154, large)-net over F₈, using

t-expansion [i] based on digital (138, 154, large)-net over F₈, using
- 7 times m-reduction [i] based on digital (138, 161, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(139, 139+15, large)-Net in Base 8 — Upper bound on s

There is no (139, 154, large)-net in base 8, because

13 times m-reduction [i] would yield (139, 141, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (139, 139+15, s)-Nets in Base 8

(139, 139+15, 2571510)-Net over F8 — Constructive and digital

(139, 139+15, large)-Net over F8 — Digital

(139, 139+15, large)-Net in Base 8 — Upper bound on s

(139, 139+15, 2571510)-Net over F₈ — Constructive and digital

(139, 139+15, large)-Net over F₈ — Digital