MinT - Best Known (60, 60+11, s)-Nets in Base 27

Best Known (60, 60+11, s)-Nets in Base 27

(60, 60+11, 1944199)-Net over F₂₇ — Constructive and digital

Digital (60, 71, 1944199)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (15, 20, 266479)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 3, 757)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
    2. digital (12, 17, 265722)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27¹⁷, 265722, F₂₇, 5, 5) (dual of [(265722, 5), 1328593, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(27¹⁷, 531445, F₂₇, 5) (dual of [531445, 531428, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(27¹⁷, 531441, F₂₇, 5) (dual of [531441, 531424, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(27¹³, 531441, F₂₇, 4) (dual of [531441, 531428, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(27⁰, 4, F₂₇, 0) (dual of [4, 4, 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]
2. digital (40, 51, 1677720)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁵¹, 1677720, F₂₇, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(27⁵¹, 8388601, F₂₇, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁵¹, large, F₂₇, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(60, 60+11, 1946802)-Net in Base 27 — Constructive

(60, 71, 1946802)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (15, 20, 269082)-net in base 27, using
  - base change [i] based on digital (10, 15, 269082)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81¹⁵, 269082, F₈₁, 5, 5) (dual of [(269082, 5), 1345395, 6]-NRT-code), using
      - appending kth column [i] based on linear OOA(81¹⁵, 269082, F₈₁, 4, 5) (dual of [(269082, 4), 1076313, 6]-NRT-code), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81⁰, s, F₈₁, 4, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81⁰, 3322, F₈₁, 4, 0) (dual of [(3322, 4), 13288, 1]-NRT-code) (see above)
        linear OOA(81¹, 3322, F₈₁, 4, 1) (dual of [(3322, 4), 13287, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81¹, s, F₈₁, 4, 1) with arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(81¹, 3322, F₈₁, 4, 1) (dual of [(3322, 4), 13287, 2]-NRT-code) (see above)
        linear OOA(81¹, 3322, F₈₁, 4, 1) (dual of [(3322, 4), 13287, 2]-NRT-code) (see above)
        linear OOA(81³, 3322, F₈₁, 4, 2) (dual of [(3322, 4), 13285, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(81³, 6643, F₈₁, 4, 2) (dual of [(6643, 4), 26569, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(81³, 6643, F₈₁, 2, 2) (dual of [(6643, 2), 13283, 3]-NRT-code), using
        appending kth column [i] based on linear OA(81³, 6643, F₈₁, 2) (dual of [6643, 6640, 3]-code), using
        Hamming code H(3,81) [i]
        linear OOA(81⁹, 3322, F₈₁, 4, 5) (dual of [(3322, 4), 13279, 6]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(81², 82, F₈₁, 4, 2) (dual of [(82, 4), 326, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;326,81) [i]
        linear OOA(81⁷, 3240, F₈₁, 4, 5) (dual of [(3240, 4), 12953, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (40, 51, 1677720)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁵¹, 1677720, F₂₇, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(27⁵¹, 8388601, F₂₇, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁵¹, large, F₂₇, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(60, 60+11, large)-Net over F₂₇ — Digital

Digital (60, 71, large)-net over F₂₇, using

5 times m-reduction [i] based on digital (60, 76, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁷⁶, large, F₂₇, 16) (dual of [large, large−76, 17]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(60, 60+11, large)-Net in Base 27 — Upper bound on s

There is no (60, 71, large)-net in base 27, because

9 times m-reduction [i] would yield (60, 62, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (60, 60+11, s)-Nets in Base 27

(60, 60+11, 1944199)-Net over F27 — Constructive and digital

(60, 60+11, 1946802)-Net in Base 27 — Constructive

(60, 60+11, large)-Net over F27 — Digital

(60, 60+11, large)-Net in Base 27 — Upper bound on s

(60, 60+11, 1944199)-Net over F₂₇ — Constructive and digital

(60, 60+11, large)-Net over F₂₇ — Digital