Construction X with Algebraic-Geometric NRT-Codes

Construction X for NRT-codes allows the construction of a new NRT-code based on two linear NRT-codes C₁ ⊂ C₂, such that the dimension of C₂ and the minimum distance of C₁ is obtained. This is bought by increasing the length of the resulting code by the length of an additional code C_e, which has to be chosen depending on the parameters of C₁ and C₂.

Given a linear [(s, T ), n₁, d₁]-code C₁, which is a subcode of a linear [(s, T ), n₂, d₂]-code C₂, as well as an additional linear [(s_e, T ), n_e, d_e]-code C_e, all over the same field, a new linear [(s + s_e, T ), n₁ + n_e, d₂ + d_e]-code can be constructed provided that n₁ + n_e ≤ n₂ and d₂ + d_e ≤ d₁. Usually C_e will be chosen such that n₂ = n₁ + n_e and d₁ = d₂ + d_e, however in some situations a smaller value of n_e or d_e may also yield good results.

Construction

Let G₁, G₂, and G_e denote the generator matrices of C₁, C₂, and C_e, respectively, such that the rows of G₁ are a subset of the rows of G₂. Let G₂ʹ denote the n_e×s matrix consisting of n_e rows of G₂ that are not in G₁. Then the new code is defined by the (n₂ + n_e)×((s + s_e), T ) generator matrix

All code words formed by a non-trivial linear combination of the first n₁ vectors have a weight of at least d₁ because these are essentially the code words of C₁ with s_eT additional 0’s appended. All other non-zero code words have a weight of at least d₂ + d_e because they are built using a non-zero code word from C₂ next to a non-zero code word from C_e.

Special Cases

• If C₁ is the [(s, T ), 0, sT + 1]-trivial code, which is a subcode of every linear code C₂, construction X reduces to juxtaposition of C₂ and C_e.

• If C₁ = C₂, the auxiliary code C_e must be an [s_e, 0, s_e + 1]-trivial code and construction X reduces to embedding C₁ in the larger space F_b^s+s_e.

Applications

MinT applies construction X in the following situations:

• Reed-Solomon NRT-codes RS(T ;n₁, b) ⊂ RS(T ;n₂, b) with n₁ < n₂ (not used)

• Algebraic-geometric NRT-codes AG(T ;F, n₁) ⊂ AG(T ;F, n₂) with n₁ < n₂

• Dual codes of OOAs from (t, s)-sequence

See Also

• For T = 1 the special case for linear codes is obtained

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 X with Algebraic-Geometric NRT-Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_OConsX-Goppa.html