MinT - Best Known (83−21, 83, s)-Nets in Base 7

Best Known (83−21, 83, s)-Nets in Base 7

(83−21, 83, 480)-Net over F₇ — Constructive and digital

Digital (62, 83, 480)-net over F₇, using

7¹ times duplication [i] based on digital (61, 82, 480)-net over F₇, using
- net defined by OOA [i] based on linear OOA(7⁸², 480, F₇, 21, 21) (dual of [(480, 21), 9998, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(7⁸², 4801, F₇, 21) (dual of [4801, 4719, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(7⁸², 4804, F₇, 21) (dual of [4804, 4722, 22]-code), using
      - trace code [i] based on linear OA(49⁴¹, 2402, F₄₉, 21) (dual of [2402, 2361, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 2402Â | 49⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(83−21, 83, 4849)-Net over F₇ — Digital

Digital (62, 83, 4849)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸³, 4849, F₇, 21) (dual of [4849, 4766, 22]-code), using
- 42 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 41 times 0) [i] based on linear OA(7⁸², 4806, F₇, 21) (dual of [4806, 4724, 22]-code), using
  - trace code [i] based on linear OA(49⁴¹, 2403, F₄₉, 21) (dual of [2403, 2362, 22]-code), using
    - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
      1. linear OA(49⁴¹, 2401, F₄₉, 21) (dual of [2401, 2360, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      2. linear OA(49³⁹, 2401, F₄₉, 20) (dual of [2401, 2362, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [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]

(83−21, 83, 6421382)-Net in Base 7 — Upper bound on s

There is no (62, 83, 6421383)-net in base 7, because

1 times m-reduction [i] would yield (62, 82, 6421383)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1986 275021 629224 117348 087603 390841 810476 140311 131790 969360 983052 141333 > 7⁸² [i]