## Juxtaposition

Give an (s1, N, d1)-code C1 and an (s2, N, d2)-code C2, an (s1 + s2, N, d1 + d2)-code over the same field can be constructed. Therefore a linear orthogonal array OA(bs1+s2−n, s1 + s2, Sb, k1 + k2 + 1) can be constructed from a linear OA(bs1n, s1, Sb, k1) and a linear OA(bs2n, s2, Sb, k2).

### Construction for Linear Codes

Let C1 = {x1,…,xN} and C2 = {y1,…yN}, then the new code C is given by

C = {(xi,yi)  :  i = 1,…, N}.

If the Ci are linear [si, n, di]-codes with generator matrices Gi, then C is an [s1 + s2, n, d1 + d2]-code with generator matrix

(.

If Gi = (InAi) and Hi = (BiImi) with Bi = – AiT and mi = sin is a parity check matrix of Ci, then the generator matrix of C is given by

()

and its parity check matrix by

see ([1, Problem 10.1]).

• Generalization for arbitrary OOAs

• Juxtaposition is a special case of construction X4

• [2, Section 1.3, Problem (17)], [2, Figure 2.6], or [1, Section 10.2].

### References

 [1] A. S. Hedayat, Neil J. A. Sloane, and John Stufken.Orthogonal Arrays.Springer Series in Statistics. Springer-Verlag, 1999. [2] F. Jessie MacWilliams and Neil J. A. Sloane.The Theory of Error-Correcting Codes.North-Holland, Amsterdam, 1977.