Linear Codes with Explicit Generator Matrix

Binary Codes

A linear [24, 14, 6]-code over 2 is given in [1]. An explicit parity check matrix can be found, e.g., in [2, Table 5.11].

Generator matrices for a linear [162, 8, 80]-code over 2 is given in [3]. A linear [159, 8, 78]-code can be obtained by discarding three columns from the generator matrix.

Ternary Codes

A linear [15, 6, 7]-code over 3 is discovered in [4]. A possible generator matrix of such a code is

$\displaystyle \left(\vphantom{\vec{I}_{6}\begin{array}{ccccccccc} 2 & 2 & 2 & 2… … 1 & 2 & 0 & 2 & 0 & 2\\ 1 & 0 & 2 & 1 & 2 & 1 & 1 & 0 & 2\end{array}}\right.$I6$\displaystyle \begin{array}{ccccccccc} 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\ 1 … … 1 & 0 & 1 & 2 & 0 & 2 & 0 & 2\\ 1 & 0 & 2 & 1 & 2 & 1 & 1 & 0 & 2\end{array}$$\displaystyle \left.\vphantom{\vec{I}_{6}\begin{array}{ccccccccc} 2 & 2 & 2 & 2… … 1 & 2 & 0 & 2 & 0 & 2\\ 1 & 0 & 2 & 1 & 2 & 1 & 1 & 0 & 2\end{array}}\right)$.

A linear [16, 5, 9]-code over 3 is discovered in [5]. A possible generator matrix of such a code is

$\displaystyle \left(\vphantom{\vec{I}_{5}\begin{array}{ccccccccccc} 2 & 2 & 2 &… … 2 & 1 & 2 & 0\\ 2 & 1 & 0 & 1 & 1 & 2 & 1 & 0 & 2 & 0 & 2\end{array}}\right.$I5$\displaystyle \begin{array}{ccccccccccc} 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 … … 0 & 0 & 2 & 1 & 2 & 0\\ 2 & 1 & 0 & 1 & 1 & 2 & 1 & 0 & 2 & 0 & 2\end{array}$$\displaystyle \left.\vphantom{\vec{I}_{5}\begin{array}{ccccccccccc} 2 & 2 & 2 &… … 2 & 1 & 2 & 0\\ 2 & 1 & 0 & 1 & 1 & 2 & 1 & 0 & 2 & 0 & 2\end{array}}\right)$.

Generator matrices for ternary [44, 6, 27]-, [76, 6, 48]-, [94, 6, 60]-, [124, 6, 81]-, [130, 6, 84]-, [134, 6, 87]-, [138, 6, 90]-, [148, 6, 96]-, [152, 6, 99]-, [156, 6, 102]-, [164, 6, 108]-, [170, 6, 111]-, [179, 6, 117]-, [188, 6, 123]-, [206, 6, 135]-, [211, 6, 138]-, [224, 6, 147]-, and [236, 6, 156]-codes are given in [6, Theorem 1(i)]; for [31, 7, 17]- and [33, 7, 18]-codes in [6, Theorem 2].

Quaternary Codes

The generator matrices for linear [28, 4, 20]-, [31, 4, 22]-, and [49, 4, 36]-codes are given in [7, Theorem 3.3].

A linear [18, 9, 8]-code over F4 is discovered in [8]. A possible generator matrix of such a code is

$\displaystyle \left(\vphantom{\vec{I}_{9}\begin{array}{ccccccccc} 1 & 1 & 1 & 1… … & \omega & 0 & 0 & \omega^{2} & \omega^{2} & \omega^{2} & 1\end{array}}\right.$I9$\displaystyle \begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ \o… … 1 & 1 & \omega & 0 & 0 & \omega^{2} & \omega^{2} & \omega^{2} & 1\end{array}$$\displaystyle \left.\vphantom{\vec{I}_{9}\begin{array}{ccccccccc} 1 & 1 & 1 & 1… … & \omega & 0 & 0 & \omega^{2} & \omega^{2} & \omega^{2} & 1\end{array}}\right)$,

with ω denoting a primitive element in F4.

The generator matrix of a linear [33, 8, 18]-code over F4 can be found in [9, Theorem 1].

Generator matrices for linear [43, 5, 30]-, [46, 5, 32]-, [51, 5, 36]-, [88, 5, 64]-, [165, 5, 122]-, [189, 5, 140]-, [219, 5, 162]-codes over F4 are listed in [10, Theorem 5 and Appendix]; [32, 5, 21]-, [57, 5, 40]-, [67, 5, 48]-, [94, 5, 68]-, [97, 5, 70]-, [126, 5, 92]-, [179, 5, 132]-, [184, 5, 136]-, and [211, 5, 156]-codes in [10, Theorem 6 and Appendix].

Boukliev, Daskalov, and Kapralov list generator matrices for linear [33, 5, 22]-, [131, 5, 96]-, [142, 5, 104]-, [147, 5, 108]-, [152, 5, 112]-, [158, 5, 116]-, [195, 5, 144]-, [200, 5, 148]-, [227, 5, 168]-, [232, 5, 172]-, [237, 5, 176]-, [242, 5, 180]-, and [247, 5, 174]-codes over F4 in [11, Theorem 11 and Appendix].

Codes over 5

[12, Theorem 4.4] lists generator matrices for linear [12, 4, 8]-, [58, 4, 45]-, [64, 4, 50]-, [76, 4, 60]-, [89, 4, 70]-, [95, 4, 75]- and [189, 4, 150]-codes over 5. The generator matrices of the first one is

$\displaystyle \left(\vphantom{\vec{I}_{4}\begin{array}{cccccccc} 0 & 1 & 1 & 1 … … 1 & 0 & 3 & 2 & 3 & 1 & 2\\ 1 & 2 & 4 & 2 & 3 & 1 & 1 & 0\end{array}}\right.$I4$\displaystyle \begin{array}{cccccccc} 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 0 &… …\\ 1 & 1 & 0 & 3 & 2 & 3 & 1 & 2\\ 1 & 2 & 4 & 2 & 3 & 1 & 1 & 0\end{array}$$\displaystyle \left.\vphantom{\vec{I}_{4}\begin{array}{cccccccc} 0 & 1 & 1 & 1 … … 1 & 0 & 3 & 2 & 3 & 1 & 2\\ 1 & 2 & 4 & 2 & 3 & 1 & 1 & 0\end{array}}\right)$.

Furthermore, a [46, 4, 35]-code is constructed in [12, Theorem 4.6].

A [12, 6, 6]-code over 5 with generator matrix

$\displaystyle \left(\vphantom{\vec{I}_{6}\begin{array}{cccccc} 1 & 1 & 1 & 1 & … … & 3 & 2\\ 4 & 1 & 2 & 1 & 3 & 2\\ 0 & 1 & 4 & 2 & 4 & 2\end{array}}\right.$I6$\displaystyle \begin{array}{cccccc} 1 & 1 & 1 & 1 & 1 & 1\\ 4 & 2 & 0 & 3 & 1… … & 4 & 3 & 3 & 2\\ 4 & 1 & 2 & 1 & 3 & 2\\ 0 & 1 & 4 & 2 & 4 & 2\end{array}$$\displaystyle \left.\vphantom{\vec{I}_{6}\begin{array}{cccccc} 1 & 1 & 1 & 1 & … … & 3 & 2\\ 4 & 1 & 2 & 1 & 3 & 2\\ 0 & 1 & 4 & 2 & 4 & 2\end{array}}\right)$

can be found in [12].

A [15, 6, 8]-code over 5 with generator matrix

$\displaystyle \left(\vphantom{\vec{I}_{6}\begin{array}{ccccccccc} 2 & 0 & 1 & 1… … 1 & 1 & 3 & 2 & 0 & 1\\ 0 & 1 & 1 & 0 & 1 & 1 & 1 & 3 & 2\end{array}}\right.$I6$\displaystyle \begin{array}{ccccccccc} 2 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 3\\ 1 … … 0 & 1 & 1 & 1 & 3 & 2 & 0 & 1\\ 0 & 1 & 1 & 0 & 1 & 1 & 1 & 3 & 2\end{array}$$\displaystyle \left.\vphantom{\vec{I}_{6}\begin{array}{ccccccccc} 2 & 0 & 1 & 1… … 1 & 1 & 3 & 2 & 0 & 1\\ 0 & 1 & 1 & 0 & 1 & 1 & 1 & 3 & 2\end{array}}\right)$

can be found in [13, Theorem 5]. The article also lists generator matrices of [18, 7, 9]-, [21, 8, 10]-, [30, 6, 20]-, [38, 8, 22]-, and [59, 8, 37]-codes over 5.

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Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Linear Codes with Explicit Generator Matrix.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CExplicitMatrix.html

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