MinT - Best Known (112−22, 112, s)-Nets in Base 8

Best Known (112−22, 112, s)-Nets in Base 8

(112−22, 112, 3007)-Net over F₈ — Constructive and digital

Digital (90, 112, 3007)-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 (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]

(112−22, 112, 5959)-Net in Base 8 — Constructive

(90, 112, 5959)-net in base 8, using

base change [i] based on digital (62, 84, 5959)-net over F₁₆, using
- 16¹ times duplication [i] based on digital (61, 83, 5959)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁸³, 5959, F₁₆, 22, 22) (dual of [(5959, 22), 131015, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(16⁸³, 65549, F₁₆, 22) (dual of [65549, 65466, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁸³, 65550, F₁₆, 22) (dual of [65550, 65467, 23]-code), using
        constructionÂ X applied to C_e(21) âŠ‚ C_e(18) [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(16⁶⁹, 65536, F₁₆, 19) (dual of [65536, 65467, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(16², 14, F₁₆, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,16)), using
        discarding factors / shortening the dual code based on linear OA(16², 16, F₁₆, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
        Reedâ€“Solomon code RS(14,16) [i]

(112−22, 112, 81270)-Net over F₈ — Digital

Digital (90, 112, 81270)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−22, 112, large)-Net in Base 8 — Upper bound on s

There is no (90, 112, large)-net in base 8, because

20 times m-reduction [i] would yield (90, 92, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (112−22, 112, s)-Nets in Base 8

(112−22, 112, 3007)-Net over F8 — Constructive and digital

(112−22, 112, 5959)-Net in Base 8 — Constructive

(112−22, 112, 81270)-Net over F8 — Digital

(112−22, 112, large)-Net in Base 8 — Upper bound on s

(112−22, 112, 3007)-Net over F₈ — Constructive and digital

(112−22, 112, 81270)-Net over F₈ — Digital