MinT - Best Known (83−21, 83, s)-Nets in Base 8

Best Known (83−21, 83, s)-Nets in Base 8

(83−21, 83, 819)-Net over F₈ — Constructive and digital

Digital (62, 83, 819)-net over F₈, using

8¹ times duplication [i] based on digital (61, 82, 819)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁸², 819, F₈, 21, 21) (dual of [(819, 21), 17117, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(8⁸², 8191, F₈, 21) (dual of [8191, 8109, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁸², 8194, F₈, 21) (dual of [8194, 8112, 22]-code), using
      - trace code [i] based on linear OA(64⁴¹, 4097, F₆₄, 21) (dual of [4097, 4056, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(83−21, 83, 8198)-Net over F₈ — Digital

Digital (62, 83, 8198)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸³, 8198, F₈, 21) (dual of [8198, 8115, 22]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(8⁸², 8196, F₈, 21) (dual of [8196, 8114, 22]-code), using
    - trace code [i] based on linear OA(64⁴¹, 4098, F₆₄, 21) (dual of [4098, 4057, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        linear OA(64⁴¹, 4096, F₆₄, 21) (dual of [4096, 4055, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(64³⁹, 4096, F₆₄, 20) (dual of [4096, 4057, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
  2. linear OA(8⁸², 8197, F₈, 20) (dual of [8197, 8115, 21]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8⁸²Â > V_b^sâˆ’1(kâˆ’1)Â = 20 954701 272470 196697 694744 755038 080263 856542 204191 735188 348862 794557 145600 [i]
  3. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 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]

(83−21, 83, large)-Net in Base 8 — Upper bound on s

There is no (62, 83, large)-net in base 8, because

19 times m-reduction [i] would yield (62, 64, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (83−21, 83, s)-Nets in Base 8

(83−21, 83, 819)-Net over F8 — Constructive and digital

(83−21, 83, 8198)-Net over F8 — Digital

(83−21, 83, large)-Net in Base 8 — Upper bound on s

(83−21, 83, 819)-Net over F₈ — Constructive and digital

(83−21, 83, 8198)-Net over F₈ — Digital