MinT - Best Known (41−12, 41, s)-Nets in Base 8

Best Known (41−12, 41, s)-Nets in Base 8

(41−12, 41, 683)-Net over F₈ — Constructive and digital

Digital (29, 41, 683)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁴¹, 683, F₈, 12, 12) (dual of [(683, 12), 8155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8⁴¹, 4098, F₈, 12) (dual of [4098, 4057, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁴¹, 4100, F₈, 12) (dual of [4100, 4059, 13]-code), using
    - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
      1. linear OA(8⁴¹, 4096, F₈, 12) (dual of [4096, 4055, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      2. linear OA(8³⁷, 4096, F₈, 11) (dual of [4096, 4059, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 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]

(41−12, 41, 2645)-Net over F₈ — Digital

Digital (29, 41, 2645)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴¹, 2645, F₈, 12) (dual of [2645, 2604, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴¹, 4096, F₈, 12) (dual of [4096, 4055, 13]-code), using
  - an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(41−12, 41, 634215)-Net in Base 8 — Upper bound on s

There is no (29, 41, 634216)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 633903 188239 392579 425092 608763 342927 > 8⁴¹ [i]