Construction XX with De Boer–Brouwer Codes
Construction XX [1] allows the construction of a new code based on linear codes C3, C1 ⊆ C3, and C2 ⊆ C3, such that the dimension of C3 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 C4 and C5, which have to be chosen depending on the parameters of the other codes.
Let C3 denote an [s, n3, d3]-code, Ci for i ∈ {1, 2} an [s, ni, di]-code contained in C3, and let C∩ := C1∩C2 have parameters [s, n∩, d∩]. Furthermore, let Ci for i ∈ {4, 5} denote codes with parameters [si, n3 – ni−3, di], all over the same field. Then a new linear [s + s4 + s5, n3, d]-code can be constructed with
Construction
Let G denote a generator matrix of C3 such that ni rows from G form generator matrices of Ci for i ∈ {1, 2,∩}. Let G4 and G5 denote generator matrices of C4 and C5, respectively.
The new generator matrix is obtained by juxtaposition of G, G4, and G5 such that the rows of G4 are aligned with the rows of G not in C1 and the rows of G5 are aligned with the rows of G not in C2. 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 C1 = C(A1), C2 = C(A2), C3 = C(A1∪A2) and C∩ = C(A1∩A2)
Codes Cr1, Dr1, Cr2, and Dr2 by de Boer and Brouwer
Special Cases
An important special case is C1 ⊆ C2. In this case C∩ = C1 and we have C1 ⊆ C2 ⊆ C3. 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: 2015-09-03.
http://mint.sbg.ac.at/desc_CConsXX-BoerBrouwer.html