MinT - Best Known (101−20, 101, s)-Nets in Base 8

Best Known (101−20, 101, s)-Nets in Base 8

(101−20, 101, 3305)-Net over F₈ — Constructive and digital

Digital (81, 101, 3305)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 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 (66, 86, 3277)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁸⁶, 3277, F₈, 20, 20) (dual of [(3277, 20), 65454, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(8⁸⁶, 32770, F₈, 20) (dual of [32770, 32684, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁸⁶, 32773, F₈, 20) (dual of [32773, 32687, 21]-code), using
        constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
        linear OA(8⁸⁶, 32768, F₈, 20) (dual of [32768, 32682, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(8⁸¹, 32768, F₈, 19) (dual of [32768, 32687, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [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]

(101−20, 101, 6555)-Net in Base 8 — Constructive

(81, 101, 6555)-net in base 8, using

8¹ times duplication [i] based on (80, 100, 6555)-net in base 8, using
- base change [i] based on digital (55, 75, 6555)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁷⁵, 6555, F₁₆, 20, 20) (dual of [(6555, 20), 131025, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(16⁷⁵, 65550, F₁₆, 20) (dual of [65550, 65475, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(16) [i] based on
        linear OA(16⁷³, 65536, F₁₆, 20) (dual of [65536, 65463, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(16⁶¹, 65536, F₁₆, 17) (dual of [65536, 65475, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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]

(101−20, 101, 71583)-Net over F₈ — Digital

Digital (81, 101, 71583)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(101−20, 101, large)-Net in Base 8 — Upper bound on s

There is no (81, 101, large)-net in base 8, because

18 times m-reduction [i] would yield (81, 83, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (101−20, 101, s)-Nets in Base 8

(101−20, 101, 3305)-Net over F8 — Constructive and digital

(101−20, 101, 6555)-Net in Base 8 — Constructive

(101−20, 101, 71583)-Net over F8 — Digital

(101−20, 101, large)-Net in Base 8 — Upper bound on s

(101−20, 101, 3305)-Net over F₈ — Constructive and digital

(101−20, 101, 71583)-Net over F₈ — Digital