## Construction XX with De Boer–Brouwer Codes

Construction XX [1] allows the construction of a new code based on linear codes C_{3}, C_{1} ⊆ C_{3}, and C_{2} ⊆ C_{3}, such that the dimension of C_{3} and the minimum distance of the smaller codes is obtained. This is bought by increasing the length of the resulting code by the length of two auxiliary codes C_{4} and C_{5}, which have to be chosen depending on the parameters of the other codes.

Let C_{3} denote an [*s*, *n*_{3}, *d*_{3}]-code, C_{i} for *i* ∈ {1, 2} an [*s*, *n*_{i}, *d*_{i}]-code contained in C_{3}, and let C_{∩} := C_{1}∩C_{2} have parameters [*s*, *n*_{∩}, *d*_{∩}]. Furthermore, let C_{i} for *i* ∈ {4, 5} denote codes with parameters [*s*_{i}, *n*_{3} – *n*_{i−3}, *d*_{i}], all over the same field. Then a new linear [*s* + *s*_{4} + *s*_{5}, *n*_{3}, *d*]-code can be constructed with

*d*= min{

*d*

_{∩},

*d*

_{1}+

*d*

_{5},

*d*

_{2}+

*d*

_{4},

*d*

_{3}+

*d*

_{4}+

*d*

_{5}}.

### Construction

Let * G* denote a generator matrix of C

_{3}such that

*n*

_{i}rows from

*form generator matrices of C*

**G**_{i}for

*i*∈ {1, 2,∩}. Let

**G**_{4}and

**G**_{5}denote generator matrices of C

_{4}and C

_{5}, respectively.

The new generator matrix is obtained by juxtaposition of * G*,

**G**_{4}, and

**G**_{5}such that the rows of

**G**_{4}are aligned with the rows of

*not in C*

**G**_{1}and the rows of

**G**_{5}are aligned with the rows of

*not in C*

**G**_{2}. Unused entries are filled with zeros.

### Applications

In addition to the cases covered by construction X and construction XX with a chain of subcodes MinT applies construction XX in the following situations:

Cyclic codes C

_{1}= C(*A*_{1}), C_{2}= C(*A*_{2}), C_{3}= C(*A*_{1}∪*A*_{2}) and C_{∩}= C(*A*_{1}∩*A*_{2})Codes C

_{r1}, D_{r1}, C_{r2}, and D_{r2}by de Boer and Brouwer

### Special Cases

An important special case is C_{1} ⊆ C_{2}. In this case C_{∩} = C_{1} and we have C_{1} ⊆ C_{2} ⊆ C_{3}. This case is handled by the separate propagation rule construction XX for a chain of codes.

### See Also

[2, Theorem 14.2]

### References

[1] | William O. Alltop. A method for extending binary linear codes. IEEE Transactions on Information Theory, 30(6):871–872, November 1984. |

[2] | Jürgen Bierbrauer.Introduction to Coding Theory.Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004. MR2079734 (2005f:94001) |

### 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. “Construction XX with De Boer–Brouwer Codes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CConsXX-BoerBrouwer.html