MinT - Best Known (105−21, 105, s)-Nets in Base 8

Best Known (105−21, 105, s)-Nets in Base 8

(105−21, 105, 3302)-Net over F₈ — Constructive and digital

Digital (84, 105, 3302)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (70, 91, 3277)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹¹, 3277, F₈, 21, 21) (dual of [(3277, 21), 68726, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(8⁹¹, 32771, F₈, 21) (dual of [32771, 32680, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁹¹, 32773, F₈, 21) (dual of [32773, 32682, 22]-code), using
        constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        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(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(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]

(105−21, 105, 6554)-Net in Base 8 — Constructive

(84, 105, 6554)-net in base 8, using

8¹ times duplication [i] based on (83, 104, 6554)-net in base 8, using
- base change [i] based on digital (57, 78, 6554)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁷⁸, 6554, F₁₆, 21, 21) (dual of [(6554, 21), 137556, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(16⁷⁸, 65541, F₁₆, 21) (dual of [65541, 65463, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁷⁸, 65545, F₁₆, 21) (dual of [65545, 65467, 22]-code), using
        constructionÂ X applied to C_e(20) âŠ‚ C_e(18) [i] based on
        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(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(16¹, 9, F₁₆, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(105−21, 105, 65388)-Net over F₈ — Digital

Digital (84, 105, 65388)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−21, 105, large)-Net in Base 8 — Upper bound on s

There is no (84, 105, large)-net in base 8, because

19 times m-reduction [i] would yield (84, 86, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (105−21, 105, s)-Nets in Base 8

(105−21, 105, 3302)-Net over F8 — Constructive and digital

(105−21, 105, 6554)-Net in Base 8 — Constructive

(105−21, 105, 65388)-Net over F8 — Digital

(105−21, 105, large)-Net in Base 8 — Upper bound on s

(105−21, 105, 3302)-Net over F₈ — Constructive and digital

(105−21, 105, 65388)-Net over F₈ — Digital