MinT - Best Known (111−20, 111, s)-Nets in Base 8

Best Known (111−20, 111, s)-Nets in Base 8

(111−20, 111, 26218)-Net over F₈ — Constructive and digital

Digital (91, 111, 26218)-net over F₈, using

8¹ times duplication [i] based on digital (90, 110, 26218)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹¹⁰, 26218, F₈, 20, 20) (dual of [(26218, 20), 524250, 21]-NRT-code), using
  - OA 10-folding and stacking [i] based on linear OA(8¹¹⁰, 262180, F₈, 20) (dual of [262180, 262070, 21]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹¹⁰, 262181, F₈, 20) (dual of [262181, 262071, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(13) [i] based on
        linear OA(8¹⁰³, 262144, F₈, 20) (dual of [262144, 262041, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(8⁷³, 262144, F₈, 14) (dual of [262144, 262071, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(8⁷, 37, F₈, 5) (dual of [37, 30, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
        a code by Danev and Olsson [i]

(111−20, 111, 262183)-Net over F₈ — Digital

Digital (91, 111, 262183)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹¹, 262183, F₈, 20) (dual of [262183, 262072, 21]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(8¹¹⁰, 262181, F₈, 20) (dual of [262181, 262071, 21]-code), using
    - constructionÂ X applied to C_e(19) âŠ‚ C_e(13) [i] based on
      1. linear OA(8¹⁰³, 262144, F₈, 20) (dual of [262144, 262041, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
      2. linear OA(8⁷³, 262144, F₈, 14) (dual of [262144, 262071, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
      3. linear OA(8⁷, 37, F₈, 5) (dual of [37, 30, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
        a code by Danev and Olsson [i]
  2. linear OA(8¹¹⁰, 262182, F₈, 19) (dual of [262182, 262072, 20]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8¹¹⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 8 709563 608667 428235 585802 693265 642469 805517 002156 386113 718666 655956 852142 114070 132769 513810 337524 [i]
  3. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 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]

(111−20, 111, large)-Net in Base 8 — Upper bound on s

There is no (91, 111, large)-net in base 8, because

18 times m-reduction [i] would yield (91, 93, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (111−20, 111, s)-Nets in Base 8

(111−20, 111, 26218)-Net over F8 — Constructive and digital

(111−20, 111, 262183)-Net over F8 — Digital

(111−20, 111, large)-Net in Base 8 — Upper bound on s

(111−20, 111, 26218)-Net over F₈ — Constructive and digital

(111−20, 111, 262183)-Net over F₈ — Digital