MinT - Best Known (50, 67, s)-Nets in Base 7

Best Known (50, 67, s)-Nets in Base 7

(50, 67, 600)-Net over F₇ — Constructive and digital

Digital (50, 67, 600)-net over F₇, using

7¹ times duplication [i] based on digital (49, 66, 600)-net over F₇, using
- net defined by OOA [i] based on linear OOA(7⁶⁶, 600, F₇, 17, 17) (dual of [(600, 17), 10134, 18]-NRT-code), using
  - OOA 8-folding and stacking with additional row [i] based on linear OA(7⁶⁶, 4801, F₇, 17) (dual of [4801, 4735, 18]-code), using
    - discarding factors / shortening the dual code based on linear OA(7⁶⁶, 4804, F₇, 17) (dual of [4804, 4738, 18]-code), using
      - trace code [i] based on linear OA(49³³, 2402, F₄₉, 17) (dual of [2402, 2369, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 2402Â | 49⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(50, 67, 4816)-Net over F₇ — Digital

Digital (50, 67, 4816)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁶⁷, 4816, F₇, 17) (dual of [4816, 4749, 18]-code), using
- 9 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 8 times 0) [i] based on linear OA(7⁶⁶, 4806, F₇, 17) (dual of [4806, 4740, 18]-code), using
  - trace code [i] based on linear OA(49³³, 2403, F₄₉, 17) (dual of [2403, 2370, 18]-code), using
    - constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
      1. linear OA(49³³, 2401, F₄₉, 17) (dual of [2401, 2368, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      2. linear OA(49³¹, 2401, F₄₉, 16) (dual of [2401, 2370, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(49⁰, 2, F₄₉, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(50, 67, 5882978)-Net in Base 7 — Upper bound on s

There is no (50, 67, 5882979)-net in base 7, because

1 times m-reduction [i] would yield (50, 66, 5882979)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 59 768274 647709 842665 085609 237213 030035 699437 791630 341201 > 7⁶⁶ [i]