MinT - Best Known (17−4, 17, s)-Nets in Base 8

Best Known (17−4, 17, s)-Nets in Base 8

(17−4, 17, 16390)-Net over F₈ — Constructive and digital

Digital (13, 17, 16390)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹⁷, 16390, F₈, 4, 4) (dual of [(16390, 4), 65543, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(8¹⁷, 16390, F₈, 3, 4) (dual of [(16390, 3), 49153, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(8¹⁷, 32780, F₈, 4) (dual of [32780, 32763, 5]-code), using
    - constructionÂ X4 applied to C_e(3) âŠ‚ C_e(1) [i] based on
      1. 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]
      2. linear OA(8⁶, 32768, F₈, 2) (dual of [32768, 32762, 3]-code), using an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
      3. 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]
      4. 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]

(17−4, 17, 34026)-Net over F₈ — Digital

Digital (13, 17, 34026)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹⁷, 34026, F₈, 4, 4) (dual of [(34026, 4), 136087, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(8¹⁷, 34026, F₈, 3, 4) (dual of [(34026, 3), 102061, 5]-NRT-code), using
  - Gilbertâ€“VarÅ¡amov bound [i]

(17−4, 17, 65280)-Net in Base 8 — Constructive

(13, 17, 65280)-net in base 8, using

8¹ times duplication [i] based on (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]

(17−4, 17, 130560)-Net in Base 8

(13, 17, 130560)-net in base 8, using

8¹ times duplication [i] based on (12, 16, 130560)-net in base 8, using
- base change [i] based on digital (8, 12, 130560)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [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]

(17−4, 17, large)-Net in Base 8 — Upper bound on s

There is no (13, 17, large)-net in base 8, because

2 times m-reduction [i] would yield (13, 15, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]