MinT - Best Known (126, 140, s)-Nets in Base 8

Best Known (126, 140, s)-Nets in Base 8

(126, 140, 2407675)-Net over F₈ — Constructive and digital

Digital (126, 140, 2407675)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (27, 34, 10933)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 9)-net over F₈, using
      - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
        the rational function field F₈(x) [i]
        Niederreiter sequence [i]
    2. digital (24, 31, 10924)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8³¹, 10924, F₈, 7, 7) (dual of [(10924, 7), 76437, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(8³¹, 32773, F₈, 7) (dual of [32773, 32742, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(8³¹, 32768, F₈, 7) (dual of [32768, 32737, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(8²⁶, 32768, F₈, 6) (dual of [32768, 32742, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 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]
2. digital (92, 106, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(126, 140, 2418588)-Net in Base 8 — Constructive

(126, 140, 2418588)-net in base 8, using

base change [i] based on digital (91, 105, 2418588)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (18, 25, 21846)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16²⁵, 21846, F₁₆, 7, 7) (dual of [(21846, 7), 152897, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(16²⁵, 65539, F₁₆, 7) (dual of [65539, 65514, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(16²⁵, 65540, F₁₆, 7) (dual of [65540, 65515, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(16²⁵, 65536, F₁₆, 7) (dual of [65536, 65511, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(16²¹, 65536, F₁₆, 6) (dual of [65536, 65515, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(16⁰, 4, F₁₆, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
  2. digital (66, 80, 2396742)-net over F₁₆, using
    - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
        OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(126, 140, large)-Net over F₈ — Digital

Digital (126, 140, large)-net over F₈, using

t-expansion [i] based on digital (125, 140, large)-net over F₈, using
- 6 times m-reduction [i] based on digital (125, 146, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴⁶, large, F₈, 21) (dual of [large, large−146, 22]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(8¹⁴⁵, large, F₈, 21) (dual of [large, large−145, 22]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(126, 140, large)-Net in Base 8 — Upper bound on s

There is no (126, 140, large)-net in base 8, because

12 times m-reduction [i] would yield (126, 128, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (126, 140, s)-Nets in Base 8

(126, 140, 2407675)-Net over F8 — Constructive and digital

(126, 140, 2418588)-Net in Base 8 — Constructive

(126, 140, large)-Net over F8 — Digital

(126, 140, large)-Net in Base 8 — Upper bound on s

(126, 140, 2407675)-Net over F₈ — Constructive and digital

(126, 140, large)-Net over F₈ — Digital