MinT - Best Known (111−19, 111, s)-Nets in Base 8

Best Known (111−19, 111, s)-Nets in Base 8

(111−19, 111, 58254)-Net over F₈ — Constructive and digital

Digital (92, 111, 58254)-net over F₈, using

8¹ times duplication [i] based on digital (91, 110, 58254)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹¹⁰, 58254, F₈, 19, 19) (dual of [(58254, 19), 1106716, 20]-NRT-code), using
  - OOA 9-folding and stacking with additional row [i] based on linear OA(8¹¹⁰, 524287, F₈, 19) (dual of [524287, 524177, 20]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹¹⁰, 524290, F₈, 19) (dual of [524290, 524180, 20]-code), using
      - trace code [i] based on linear OA(64⁵⁵, 262145, F₆₄, 19) (dual of [262145, 262090, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(111−19, 111, 524296)-Net over F₈ — Digital

Digital (92, 111, 524296)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹¹, 524296, F₈, 19) (dual of [524296, 524185, 20]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(8¹¹⁰, 524294, F₈, 19) (dual of [524294, 524184, 20]-code), using
    - trace code [i] based on linear OA(64⁵⁵, 262147, F₆₄, 19) (dual of [262147, 262092, 20]-code), using
      - constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
        linear OA(64⁵⁵, 262144, F₆₄, 19) (dual of [262144, 262089, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(64⁵², 262144, F₆₄, 18) (dual of [262144, 262092, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
  2. linear OA(8¹¹⁰, 524295, F₈, 18) (dual of [524295, 524185, 19]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8¹¹⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 11 175336 576496 700365 290371 242454 314283 012677 675582 594754 438476 389156 054738 904322 563758 966076 899328 [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]

(111−19, 111, large)-Net in Base 8 — Upper bound on s

There is no (92, 111, large)-net in base 8, because

17 times m-reduction [i] would yield (92, 94, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (111−19, 111, s)-Nets in Base 8

(111−19, 111, 58254)-Net over F8 — Constructive and digital

(111−19, 111, 524296)-Net over F8 — Digital

(111−19, 111, large)-Net in Base 8 — Upper bound on s

(111−19, 111, 58254)-Net over F₈ — Constructive and digital

(111−19, 111, 524296)-Net over F₈ — Digital