MinT - Best Known (82−23, 82, s)-Nets in Base 7

Best Known (82−23, 82, s)-Nets in Base 7

(82−23, 82, 300)-Net over F₇ — Constructive and digital

Digital (59, 82, 300)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 14, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 7, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
      - Niederreiter sequence [i]
2. digital (11, 22, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 11, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
3. digital (23, 46, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 23, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(82−23, 82, 2447)-Net over F₇ — Digital

Digital (59, 82, 2447)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸², 2447, F₇, 23) (dual of [2447, 2365, 24]-code), using
- 37 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 1, 8 times 0, 1, 23 times 0) [i] based on linear OA(7⁷⁷, 2405, F₇, 23) (dual of [2405, 2328, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
    1. linear OA(7⁷⁷, 2401, F₇, 23) (dual of [2401, 2324, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(7⁷³, 2401, F₇, 22) (dual of [2401, 2328, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(7⁰, 4, F₇, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(82−23, 82, 1367273)-Net in Base 7 — Upper bound on s

There is no (59, 82, 1367274)-net in base 7, because

1 times m-reduction [i] would yield (59, 81, 1367274)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 283 754671 838140 559534 322817 044270 481342 826245 779865 206999 483294 267385 > 7⁸¹ [i]