Concatenation of Two NRT-Codes

Given a linear [(s1, T1), n1, d1]-NRT-code C1 over Fbn2 and a linear [(s2, T2), n2, d2]-NRT-code C2 over Fb, a linear [(s1s2, T1T2), n1n2, d1d2]-NRT-code over Fb can be constructed. Using duality, a linear ordered orthogonal array OOA(bs1s2T1T2-n1n2, s1s2,Fb, T1T2, d1d2 − 1) can be constructed based on a linear OOA(bs1T1-m1, s1,Fbs2T2-n2, d1 − 1) and OOA(bs2T2-m2, s2,Fb, T2, d2 − 1).

Construction

Let φ : Fbn2↔C2 ⊆ Fb(s2, T2) denote an arbitrary Fb-linear bijection. Then the new code C is defined as

C := {(φ(xi))(i, j) ∈ {1,…, s1}×{1,…, T1}  :  x ∈ C1}

with the resulting indices for depth ordered first by j and secondly by the depth-index from C2. More formally and assuming that the columns of Ch are indexed by {0,…, sh – 1}×{0,…, Th – 1} for h = 1, 2, C is defined as

C := {((φ(x⌊i/s2⌋,⌊j/T2⌋))i mod s2, j mod T2)(i, j) ∈ {0,…, s1s2−1}×{0,…, T1T2−1}  :  x ∈ C1}.

See Also

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. “Concatenation of Two NRT-Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_OConcatD.html

Show usage of this method