MinT - Best Known (73, 82, s)-Nets in Base 8

Best Known (73, 82, s)-Nets in Base 8

(73, 82, 4210686)-Net over F₈ — Constructive and digital

Digital (73, 82, 4210686)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (12, 16, 16386)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁶, 16386, F₈, 4, 4) (dual of [(16386, 4), 65528, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹⁶, 16386, F₈, 3, 4) (dual of [(16386, 3), 49142, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8¹⁶, 32772, F₈, 4) (dual of [32772, 32756, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹⁶, 32773, F₈, 4) (dual of [32773, 32757, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(8¹⁶, 32768, F₈, 4) (dual of [32768, 32752, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(8¹¹, 32768, F₈, 3) (dual of [32768, 32757, 4]-code or 32768-cap in PG(10,8)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 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 (57, 66, 4194300)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁶⁶, 4194300, F₈, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁶⁶, 8388601, F₈, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁶⁶, 8388602, F₈, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
        trace code [i] based on linear OOA(64³³, 4194301, F₆₄, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64³³, 8388602, F₆₄, 9) (dual of [8388602, 8388569, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(64³³, large, F₆₄, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(73, 82, 4259580)-Net in Base 8 — Constructive

(73, 82, 4259580)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (12, 16, 65280)-net in base 8, using
  - base change [i] based on digital (8, 12, 65280)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16¹², 65280, F₁₆, 4, 4) (dual of [(65280, 4), 261108, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(16¹², 130560, F₁₆, 4) (dual of [130560, 130548, 5]-code), using
        trace code [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (57, 66, 4194300)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁶⁶, 4194300, F₈, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁶⁶, 8388601, F₈, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁶⁶, 8388602, F₈, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
        trace code [i] based on linear OOA(64³³, 4194301, F₆₄, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64³³, 8388602, F₆₄, 9) (dual of [8388602, 8388569, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(64³³, large, F₆₄, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(73, 82, large)-Net over F₈ — Digital

Digital (73, 82, large)-net over F₈, using

8¹ times duplication [i] based on digital (72, 81, large)-net over F₈, using
- t-expansion [i] based on digital (69, 81, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸¹, large, F₈, 12) (dual of [large, large−81, 13]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(73, 82, large)-Net in Base 8 — Upper bound on s

There is no (73, 82, large)-net in base 8, because

7 times m-reduction [i] would yield (73, 75, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (73, 82, s)-Nets in Base 8

(73, 82, 4210686)-Net over F8 — Constructive and digital

(73, 82, 4259580)-Net in Base 8 — Constructive

(73, 82, large)-Net over F8 — Digital

(73, 82, large)-Net in Base 8 — Upper bound on s

(73, 82, 4210686)-Net over F₈ — Constructive and digital

(73, 82, large)-Net over F₈ — Digital