Construction XX with a Chain of Algebraic-Geometric Codes

Given three linear [s, n_i, d_i]-codes C₁ ⊆ C₂ ⊆ C₃ and two additional codes C₄, C₅ with parameters [s₄, n₃ – n₁, d₄] and [s₅, n₃ – n₂, d₅], all over the same field, a new [s + s₄ + s₅, n₃, d]-code with

can be constructed.

This construction is a special case of construction XX, which requires only C₁ ⊆ C₃ instead of C₁ ⊆ C₂.

Special Cases

• If C₁ = C₂, this construction yields the same result as construction X applied to C₁ ⊆ C₃ with the extension code obtained from applying juxtaposition to C₄ and C₅.

• If C₂ = C₃, this construction yields the same result as construction X applied to C₁ ⊆ C₂ with extension code C₅ and increasing the length of the code by s₅ using embedding.

Applications

MinT applies this method in all situations where construction X is applied and chains containing at least 3 subcodes are available.

• Reed-Solomon-codes RS(n₁, b) ⊂ RS(n₂, b) ⊂ RS(n₃, b) with n₁ < n₂ < n₃ (not used)

• Algebraic-geometric codes AG(F, n₁) ⊂ AG(F, n₂) ⊂ AG(F, n₃) with n₁ < n₂ < n₃

• Cyclic codes C(A₁) ⊂ C(A₂) ⊂ C(A₃) with A₃ʹ ⊂ A₂ʹ ⊂ A₁ʹ

• Extended cyclic codes C_e(k₁) ⊂ C_e(k₂) ⊂ C_e(k₃) with k₁ > k₂ > k₃

• Sequences C_r and D_r by de Boer and Brouwer (not used)

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 XX with a Chain of Algebraic-Geometric Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CConsXX3-Goppa.html