## Augmented Codes by De Boer and Brouwer

In  and  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

[ , 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

• 2u−1 – 2u−r−2 − 1 if b = 2 and u is odd,

• 2u−1 – 2u−r−1 − 1 if b = 2 and u is even, and

• 3u−1 – (3u−r−1 + 1)/2 if b = 3.

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 = { 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

  Mario A. de Boer.Codes: Their Parameters and Geometry.PhD thesis, Eindhoven Univ. Techn., 1997.  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)