MinT - Best Known (123−12, 123, s)-Nets in Base 8

Best Known (123−12, 123, s)-Nets in Base 8

(123−12, 123, 2970965)-Net over F₈ — Constructive and digital

Digital (111, 123, 2970965)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (27, 33, 174765)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8³³, 174765, F₈, 6, 6) (dual of [(174765, 6), 1048557, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(8³³, 524295, F₈, 6) (dual of [524295, 524262, 7]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(8³², 524294, F₈, 6) (dual of [524294, 524262, 7]-code), using
        trace code [i] based on linear OA(64¹⁶, 262147, F₆₄, 6) (dual of [262147, 262131, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(64¹⁶, 262144, F₆₄, 6) (dual of [262144, 262128, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(64¹³, 262144, F₆₄, 5) (dual of [262144, 262131, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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. digital (78, 90, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁰, 2796200, F₈, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁰, 8388601, F₈, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁰, 8388602, F₈, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁵, 4194301, F₆₄, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁵, 8388602, F₆₄, 12) (dual of [8388602, 8388557, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(123−12, 123, large)-Net over F₈ — Digital

Digital (111, 123, large)-net over F₈, using

t-expansion [i] based on digital (106, 123, large)-net over F₈, using
- 1 times m-reduction [i] based on digital (106, 124, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²⁴, large, F₈, 18) (dual of [large, large−124, 19]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(8¹²¹, large, F₈, 18) (dual of [large, large−121, 19]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(123−12, 123, large)-Net in Base 8 — Upper bound on s

There is no (111, 123, large)-net in base 8, because

10 times m-reduction [i] would yield (111, 113, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (123−12, 123, s)-Nets in Base 8

(123−12, 123, 2970965)-Net over F8 — Constructive and digital

(123−12, 123, large)-Net over F8 — Digital

(123−12, 123, large)-Net in Base 8 — Upper bound on s

(123−12, 123, 2970965)-Net over F₈ — Constructive and digital

(123−12, 123, large)-Net over F₈ — Digital