MinT - Best Known (158−23, 158, s)-Nets in Base 8

Best Known (158−23, 158, s)-Nets in Base 8

(158−23, 158, 190679)-Net over F₈ — Constructive and digital

Digital (135, 158, 190679)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 16, 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 (119, 142, 190651)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴², 190651, F₈, 23, 23) (dual of [(190651, 23), 4384831, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(8¹⁴², 2097162, F₈, 23) (dual of [2097162, 2097020, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁴², 2097168, F₈, 23) (dual of [2097168, 2097026, 24]-code), using
        constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
        linear OA(8¹⁴¹, 2097153, F₈, 23) (dual of [2097153, 2097012, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 8¹⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        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(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]

(158−23, 158, 381300)-Net in Base 8 — Constructive

(135, 158, 381300)-net in base 8, using

net defined by OOA [i] based on OOA(8¹⁵⁸, 381300, S₈, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(8¹⁵⁸, 4194301, S₈, 23), using
  - discarding factors based on OA(8¹⁵⁸, 4194310, S₈, 23), using
    - trace code [i] based on OA(64⁷⁹, 2097155, S₆₄, 23), using
      - discarding parts of the base [i] based on linear OA(128⁶⁷, 2097155, F₁₂₈, 23) (dual of [2097155, 2097088, 24]-code), using
        constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
        linear OA(128⁶⁷, 2097152, F₁₂₈, 23) (dual of [2097152, 2097085, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(128⁶⁴, 2097152, F₁₂₈, 22) (dual of [2097152, 2097088, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(158−23, 158, 3958989)-Net over F₈ — Digital

Digital (135, 158, 3958989)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−23, 158, large)-Net in Base 8 — Upper bound on s

There is no (135, 158, large)-net in base 8, because

21 times m-reduction [i] would yield (135, 137, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (158−23, 158, s)-Nets in Base 8

(158−23, 158, 190679)-Net over F8 — Constructive and digital

(158−23, 158, 381300)-Net in Base 8 — Constructive

(158−23, 158, 3958989)-Net over F8 — Digital

(158−23, 158, large)-Net in Base 8 — Upper bound on s

(158−23, 158, 190679)-Net over F₈ — Constructive and digital

(158−23, 158, 3958989)-Net over F₈ — Digital