MinT - Best Known (143−21, 143, s)-Nets in Base 8

Best Known (143−21, 143, s)-Nets in Base 8

(143−21, 143, 209744)-Net over F₈ — Constructive and digital

Digital (122, 143, 209744)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (107, 128, 209716)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²⁸, 209716, F₈, 21, 21) (dual of [(209716, 21), 4403908, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(8¹²⁸, 2097161, F₈, 21) (dual of [2097161, 2097033, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹²⁸, 2097168, F₈, 21) (dual of [2097168, 2097040, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(8¹²⁷, 2097153, F₈, 21) (dual of [2097153, 2097026, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 8¹⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(8¹¹³, 2097153, F₈, 19) (dual of [2097153, 2097040, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 8¹⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(8¹, 15, F₈, 1) (dual of [15, 14, 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]

(143−21, 143, 209748)-Net in Base 8 — Constructive

(122, 143, 209748)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (6, 16, 33)-net in base 8, using
  - base change [i] based on digital (2, 12, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (106, 127, 209715)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²⁷, 209715, F₈, 21, 21) (dual of [(209715, 21), 4403888, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(8¹²⁷, 2097151, F₈, 21) (dual of [2097151, 2097024, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹²⁷, 2097152, F₈, 21) (dual of [2097152, 2097025, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(143−21, 143, 3398622)-Net over F₈ — Digital

Digital (122, 143, 3398622)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(143−21, 143, large)-Net in Base 8 — Upper bound on s

There is no (122, 143, large)-net in base 8, because

19 times m-reduction [i] would yield (122, 124, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (143−21, 143, s)-Nets in Base 8

(143−21, 143, 209744)-Net over F8 — Constructive and digital

(143−21, 143, 209748)-Net in Base 8 — Constructive

(143−21, 143, 3398622)-Net over F8 — Digital

(143−21, 143, large)-Net in Base 8 — Upper bound on s

(143−21, 143, 209744)-Net over F₈ — Constructive and digital

(143−21, 143, 3398622)-Net over F₈ — Digital