## Standard Lengthening for NRT-Codes

If an [s, n, d]-code C0 contains an [s, n−1, d + 1]-subcode C1 (as it is the case for Reed-Solomon codes, algebraic-geometric codes and other families of codes), C0 can be lengthened to an [s + 1, n, d + 1]-code C. If G0 is a generator matrix of C0 such that the first n−1 rows of G0 are a generator matrix of C1, a generator matrix of C is given by

G := G0.

The same result can be obtained by applying construction X to C1C0 with the [1, 1, 1]-code without redundancy.

For NRT-codes with depth T > 1 and n > 1 a similar construction is possible. Let C0 denote a linear [(s, T ), n, d0]-code. Let T ʹ := min{T , n} and let C0 contain a chain of subcodes Ci for i = 1,…, T ʹ−1 with CTʹ−1 ⊂ ⋯ ⊂ C1C0, dimensions dimCi = ni, and minimum distances d (Ci) = di. Furthermore assume that G0 denote a generator matrix of C0 such that the first ni rows of G form a generator matrix of Ci. Then C0 can be lengthened to an [(s + 1, T ), n, d]-code with

d = di + i

and generator matrix

G := G0

for nT and

G := G0 |  0n×(T-n)In

for nT .

Note that this result cannot be obtained using construction X in a straightforward way! It is only possible because the extension code Ce defined by the generator matrix I is an unequal error protection (UEP) NRT-code: If Di denotes the code defined by the first i rows of I, then d (Ce ∖ Di) = i + 1 for i = 0,…, T ʹ−1 (cf. [1]).

### References

 [1] Jürgen Bierbrauer, Yves Edel, and Ludo Tolhuizen.New codes via the lengthening of BCH codes with UEP codes.Finite Fields and Their Applications, 5(4):345–353, October 1999.doi:10.1006/ffta.1999.0245 MR1711841 (2000j:94027)