MinT - Best Known (133, 148, s)-Nets in Base 8

Best Known (133, 148, s)-Nets in Base 8

(133, 148, 2407675)-Net over F₈ — Constructive and digital

Digital (133, 148, 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 (99, 114, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(133, 148, 2418588)-Net in Base 8 — Constructive

(133, 148, 2418588)-net in base 8, using

base change [i] based on digital (96, 111, 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 (71, 86, 2396742)-net over F₁₆, using
    - trace code for nets [i] based on digital (28, 43, 1198371)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁴³, 1198371, F₂₅₆, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
        OOA 7-folding and stacking with additional row [i] based on linear OA(256⁴³, 8388598, F₂₅₆, 15) (dual of [8388598, 8388555, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴³, large, F₂₅₆, 15) (dual of [large, large−43, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(133, 148, large)-Net over F₈ — Digital

Digital (133, 148, large)-net over F₈, using

t-expansion [i] based on digital (131, 148, large)-net over F₈, using
- 5 times m-reduction [i] based on digital (131, 153, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵³, large, F₈, 22) (dual of [large, large−153, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(133, 148, large)-Net in Base 8 — Upper bound on s

There is no (133, 148, large)-net in base 8, because

13 times m-reduction [i] would yield (133, 135, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (133, 148, s)-Nets in Base 8

(133, 148, 2407675)-Net over F8 — Constructive and digital

(133, 148, 2418588)-Net in Base 8 — Constructive

(133, 148, large)-Net over F8 — Digital

(133, 148, large)-Net in Base 8 — Upper bound on s

(133, 148, 2407675)-Net over F₈ — Constructive and digital

(133, 148, large)-Net over F₈ — Digital