MinT - Best Known (109−22, 109, s)-Nets in Base 8

Best Known (109−22, 109, s)-Nets in Base 8

(109−22, 109, 2996)-Net over F₈ — Constructive and digital

Digital (87, 109, 2996)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (74, 96, 2979)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁶, 2979, F₈, 22, 22) (dual of [(2979, 22), 65442, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(8⁹⁶, 32769, F₈, 22) (dual of [32769, 32673, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁹⁶, 32773, F₈, 22) (dual of [32773, 32677, 23]-code), using
        constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
        linear OA(8⁹⁶, 32768, F₈, 22) (dual of [32768, 32672, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(8⁹¹, 32768, F₈, 21) (dual of [32768, 32677, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(109−22, 109, 5958)-Net in Base 8 — Constructive

(87, 109, 5958)-net in base 8, using

8¹ times duplication [i] based on (86, 108, 5958)-net in base 8, using
- base change [i] based on digital (59, 81, 5958)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁸¹, 5958, F₁₆, 22, 22) (dual of [(5958, 22), 130995, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(16⁸¹, 65538, F₁₆, 22) (dual of [65538, 65457, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁸¹, 65540, F₁₆, 22) (dual of [65540, 65459, 23]-code), using
        constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
        linear OA(16⁸¹, 65536, F₁₆, 22) (dual of [65536, 65455, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(16⁷⁷, 65536, F₁₆, 21) (dual of [65536, 65459, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(109−22, 109, 60386)-Net over F₈ — Digital

Digital (87, 109, 60386)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−22, 109, large)-Net in Base 8 — Upper bound on s

There is no (87, 109, large)-net in base 8, because

20 times m-reduction [i] would yield (87, 89, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (109−22, 109, s)-Nets in Base 8

(109−22, 109, 2996)-Net over F8 — Constructive and digital

(109−22, 109, 5958)-Net in Base 8 — Constructive

(109−22, 109, 60386)-Net over F8 — Digital

(109−22, 109, large)-Net in Base 8 — Upper bound on s

(109−22, 109, 2996)-Net over F₈ — Constructive and digital

(109−22, 109, 60386)-Net over F₈ — Digital