## 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 ≤ *r* ≤ *u*/2. Then there exist linear codes C_{r} over **F**_{b} with parameters

*ur*,

*b*

^{u−1}–

*b*

^{u−r−2}].

If *r* = (*u*−1)/2, the minimum distance of C_{r} is actually *b*^{u−1}, if *r* = *u*/2, it is *b*^{u−1} + *b*^{u/2–1}. Furthermore the codes are nested, namely C_{r} ⊆ C_{rʹ} for *r* ≥ *r*ʹ.

### Augmenting these Codes

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

2

^{u−1}– 2^{u−r−2}− 1 if*b*= 2 and*u*is odd,2

^{u−1}– 2^{u−r−1}− 1 if*b*= 2 and*u*is even, and3

^{u−1}– (3^{u−r−1}+ 1)/2 if*b*= 3.

Since C_{r} ⊆ C_{rʹ}, D_{r} ⊆ D_{rʹ}, and C_{r} ⊆ D_{r} for all 0 ≤ *r*ʹ ≤ *r* ≤ *u*/2, there are many possibilities for applying construction X.

### Construction

For given *b*, *u* ≥ 1, and 0 ≤ *r* ≤ *u*/2 let *F*_{r} denote the vector space of functions on **F**_{bu} defined by

*F*

_{r}= {

*a*

_{ij}

*x*

^{bi}

*x*

^{bj}:

*a*

_{ij}∈

**F**

_{bu},

*a*

_{i+1,j+1}=

*a*

_{ij}

^{b}, and

*a*

_{ij}= 0 for

*j*–

*i*<

*r*}

where *i* + 1 and *j* + 1 are calculated modulo *u*. If **F**_{bu} is identified with **F**_{b}^{u}, then *F*_{0} is the set of all (homogenous) quadratic forms on **F**_{b}^{u} and *F*_{r} for *r* > 0 defines certain subspaces of *F*_{0}. The code C_{r} is obtained from *F*_{r} as

_{r}= {(

*f*(

*)*

**x**_{x ∈ PG(u−1, b)}:

*f*∈

*F*

_{r}},

i.e., each function *f* ∈ *F*_{r} defines a code word of C_{r} 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: 2015-09-03.
http://mint.sbg.ac.at/desc_CBoerBrouwer.html