MinT - Best Known (11, 11+11, s)-Nets in Base 27

Best Known (11, 11+11, s)-Nets in Base 27

(11, 11+11, 146)-Net over F₂₇ — Constructive and digital

Digital (11, 22, 146)-net over F₂₇, using

27¹ times duplication [i] based on digital (10, 21, 146)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27²¹, 146, F₂₇, 11, 11) (dual of [(146, 11), 1585, 12]-NRT-code), using
  - OOA 5-folding and stacking with additional row [i] based on linear OA(27²¹, 731, F₂₇, 11) (dual of [731, 710, 12]-code), using
    - constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
      1. linear OA(27²¹, 729, F₂₇, 11) (dual of [729, 708, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      2. linear OA(27¹⁹, 729, F₂₇, 10) (dual of [729, 710, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
      3. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(11, 11+11, 164)-Net in Base 27 — Constructive

(11, 22, 164)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 351)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁷, 351, F₂₇, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
      - a code by Danev and Olsson [i]
2. (4, 15, 82)-net in base 27, using
  - 1 times m-reduction [i] based on (4, 16, 82)-net in base 27, using
    - base change [i] based on digital (0, 12, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]

(11, 11+11, 367)-Net over F₂₇ — Digital

Digital (11, 22, 367)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(27²², 367, F₂₇, 2, 11) (dual of [(367, 2), 712, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(27²², 734, F₂₇, 11) (dual of [734, 712, 12]-code), using
  - discarding factors / shortening the dual code based on linear OA(27²², 735, F₂₇, 11) (dual of [735, 713, 12]-code), using
    - constructionÂ X applied to C([0,5]) âŠ‚ C([0,4]) [i] based on
      1. linear OA(27²¹, 730, F₂₇, 11) (dual of [730, 709, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
      2. linear OA(27¹⁷, 730, F₂₇, 9) (dual of [730, 713, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
      3. linear OA(27¹, 5, F₂₇, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(11, 11+11, 102939)-Net in Base 27 — Upper bound on s

There is no (11, 22, 102940)-net in base 27, because

1 times m-reduction [i] would yield (11, 21, 102940)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 144596 881523 403261 658532 604089 > 27²¹ [i]