Augmented 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 ≥ rʹ.

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, and

• 3^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 ≤ rʹ ≤ 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_b^u defined by

F_r = {a_ijx^bix^bj  :  a_ij ∈ F_b^u, 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_b^u is identified with F_b^u, then F₀ is the set of all (homogenous) quadratic forms on F_b^u and F_r for r > 0 defines certain subspaces of F₀. The code C_r is obtained from F_r as

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

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. “Augmented Codes by De Boer and Brouwer.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CBoerBrouwer-augmented.html