MinT - Best Known (167−17, 167, s)-Nets in Base 8

Best Known (167−17, 167, s)-Nets in Base 8

(167−17, 167, 2105345)-Net over F₈ — Constructive and digital

Digital (150, 167, 2105345)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (29, 37, 8195)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8³⁷, 8195, F₈, 8, 8) (dual of [(8195, 8), 65523, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(8³⁷, 32780, F₈, 8) (dual of [32780, 32743, 9]-code), using
      - constructionÂ X4 applied to C_e(8) âŠ‚ C_e(5) [i] based on
        linear OA(8³⁶, 32768, F₈, 9) (dual of [32768, 32732, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(8²⁶, 32768, F₈, 6) (dual of [32768, 32742, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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 (113, 130, 2097150)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³⁰, 2097150, F₈, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹³⁰, 8388601, F₈, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹³⁰, 8388602, F₈, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
        trace code [i] based on linear OOA(64⁶⁵, 4194301, F₆₄, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶⁵, 8388602, F₆₄, 17) (dual of [8388602, 8388537, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(167−17, 167, large)-Net over F₈ — Digital

Digital (150, 167, large)-net over F₈, using

t-expansion [i] based on digital (144, 167, large)-net over F₈, using
- 1 times m-reduction [i] based on digital (144, 168, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(167−17, 167, large)-Net in Base 8 — Upper bound on s

There is no (150, 167, large)-net in base 8, because

15 times m-reduction [i] would yield (150, 152, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (167−17, 167, s)-Nets in Base 8

(167−17, 167, 2105345)-Net over F8 — Constructive and digital

(167−17, 167, large)-Net over F8 — Digital

(167−17, 167, large)-Net in Base 8 — Upper bound on s

(167−17, 167, 2105345)-Net over F₈ — Constructive and digital

(167−17, 167, large)-Net over F₈ — Digital