MinT - Best Known (67−19, 67, s)-Nets in Base 8

Best Known (67−19, 67, s)-Nets in Base 8

(67−19, 67, 456)-Net over F₈ — Constructive and digital

Digital (48, 67, 456)-net over F₈, using

8¹ times duplication [i] based on digital (47, 66, 456)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁶⁶, 456, F₈, 19, 19) (dual of [(456, 19), 8598, 20]-NRT-code), using
  - OOA 9-folding and stacking with additional row [i] based on linear OA(8⁶⁶, 4105, F₈, 19) (dual of [4105, 4039, 20]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁶⁶, 4106, F₈, 19) (dual of [4106, 4040, 20]-code), using
      - constructionÂ X applied to C([0,9]) âŠ‚ C([0,8]) [i] based on
        linear OA(8⁶⁵, 4097, F₈, 19) (dual of [4097, 4032, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(8⁵⁷, 4097, F₈, 17) (dual of [4097, 4040, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(67−19, 67, 576)-Net in Base 8 — Constructive

(48, 67, 576)-net in base 8, using

1 times m-reduction [i] based on (48, 68, 576)-net in base 8, using
- trace code for nets [i] based on (14, 34, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (14, 35, 288)-net in base 64, using
    - base change [i] based on digital (9, 30, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(67−19, 67, 3278)-Net over F₈ — Digital

Digital (48, 67, 3278)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁷, 3278, F₈, 19) (dual of [3278, 3211, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁶⁷, 4107, F₈, 19) (dual of [4107, 4040, 20]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8⁶⁶, 4106, F₈, 19) (dual of [4106, 4040, 20]-code), using
    - constructionÂ X applied to C([0,9]) âŠ‚ C([0,8]) [i] based on
      1. linear OA(8⁶⁵, 4097, F₈, 19) (dual of [4097, 4032, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      2. linear OA(8⁵⁷, 4097, F₈, 17) (dual of [4097, 4040, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
      3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(67−19, 67, 2484919)-Net in Base 8 — Upper bound on s

There is no (48, 67, 2484920)-net in base 8, because

1 times m-reduction [i] would yield (48, 66, 2484920)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 401734 699511 635562 609021 654214 272563 690881 116172 027961 421774 > 8⁶⁶ [i]