Codes by De Boer and Brouwer

In [1] and [2] the following codes are constructed: Let b denote a prime power and u ≥ 1 an integer. Let r denote an integer with 0 ≤ ru/2. Then there exist linear codes Cr over Fb with parameters

[$\displaystyle {\frac{{b^{u}−1}}{{b−1}}}$,$\displaystyle \binom{u+1}{2}$ur, bu−1bu−r−2].

If r = (u−1)/2, the minimum distance of Cr is actually bu−1, if r = u/2, it is bu−1 + bu/2–1. Furthermore the codes are nested, namely CrC for rrʹ.

Augmenting these Codes

Since the all-one vector is not a code word of any of these codes, augmented codes Dr can be obtained from Cr by appending the all-one vector to its generator matrix. Obviously Dr has the same length and a dimension increased by one compared to Cr. Furthermore it can be shown that the minimum distance of Dr is greater or equal to

Since CrC, DrD, and CrDr for all 0 ≤ rʹ ≤ ru/2, there are many possibilities for applying construction X.

Construction

For given b, u ≥ 1, and 0 ≤ ru/2 let Fr denote the vector space of functions on Fbu defined by

Fr = {$\displaystyle \sum_{{0\le i\le j<u−1}}^{}$aijxbixbj  :  aijFbu, ai+1,j+1 = aijb, and aij = 0 for ji < r}

where i + 1 and j + 1 are calculated modulo u. If Fbu is identified with Fbu, then F0 is the set of all (homogenous) quadratic forms on Fbu and Fr for r > 0 defines certain subspaces of F0. The code Cr is obtained from Fr as

Cr = {(f (x)x ∈ PG(u−1, b)  :  fFr},

i.e., each function fFr defines a code word of Cr which is obtained by applying f to representatives of all points in the projective space PG(u−1, b).

References

[1]Mario A. de Boer.
Codes: Their Parameters and Geometry.
PhD thesis, Eindhoven Univ. Techn., 1997.
[2]Andries E. Brouwer.
Linear spaces of quadrics and new good codes.
Bulletin of the Belgian Mathematical Society. Simon Stevin, 5(2–3):177–180, 1998.
MR1630022 (99j:94073)

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. “Codes by De Boer and Brouwer.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CBoerBrouwer.html

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