MinT - Best Known (87−23, 87, s)-Nets in Base 7

Best Known (87−23, 87, s)-Nets in Base 7

(87−23, 87, 344)-Net over F₇ — Constructive and digital

Digital (64, 87, 344)-net over F₇, using

base reduction for projective spaces (embedding PG(43,49) in PG(86,7)) for nets [i] based on digital (21, 44, 344)-net over F₄₉, using
- net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
    - the Hermitian function field over F₄₉ [i]

(87−23, 87, 3329)-Net over F₇ — Digital

Digital (64, 87, 3329)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸⁷, 3329, F₇, 23) (dual of [3329, 3242, 24]-code), using
- 914 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 57 times 0, 1, 115 times 0, 1, 186 times 0, 1, 239 times 0, 1, 275 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]

(87−23, 87, 3311251)-Net in Base 7 — Upper bound on s

There is no (64, 87, 3311252)-net in base 7, because

1 times m-reduction [i] would yield (64, 86, 3311252)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 4 769056 466971 706996 292639 729743 155690 167200 318080 808944 162425 612818 806177 > 7⁸⁶ [i]